Question: What year was the idea for this system developed?

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

Answer: 1637

<hl> The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. <hl> Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The idea of this system was developed in <hl> 1637 <hl> in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.


Question: Did Fermat publish the discovery?

Answer: did not

<hl> The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. <hl> Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and <hl> did not <hl> publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.


Question: Who's writings helped develop this system independent of Descartes?

Answer: Pierre de Fermat

<hl> The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat , although Fermat also worked in three dimensions and did not publish the discovery. <hl> Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The idea of this system was developed in 1637 in writings by Descartes and independently by <hl> Pierre de Fermat <hl>, although Fermat also worked in three dimensions and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The idea of this system was developed in 1637 in writings by Descartes and independently by <hl> Pierre de Fermat <hl> , although Fermat also worked in three dimensions and did not publish the discovery.


Question: Who introduced the concept by translating Descartes' works into Latin?

Answer: Frans van Schooten and his students

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. <hl> The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students . <hl> These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by <hl> Frans van Schooten and his students <hl>. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by <hl> Frans van Schooten and his students <hl> .


Question: What year did Frans van Schooten and his students translate the work into latin?

Answer: 1649

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. <hl> The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. <hl> These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in <hl> 1649 <hl> by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.


Question: How is a point that is a Cartesian coordinate usually written?

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7).  The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

Answer: in parentheses and separated by commas

<hl> The Cartesian coordinates of a point are usually written in parentheses and separated by commas , as in (10, 5) or (3, 5, 7). <hl> The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

The Cartesian coordinates of a point are usually written <hl> in parentheses and separated by commas <hl>, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

The Cartesian coordinates of a point are usually written <hl> in parentheses and separated by commas <hl> , as in (10, 5) or (3, 5, 7).


Question: What is labelled with the capital letter O?

Answer: The origin

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). <hl> The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. <hl> This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). <hl> The origin <hl> is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.


Question: How are unknown coordinates denoted?

Answer: by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). <hl> The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space . <hl> This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted <hl> by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space <hl>. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted <hl> by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space <hl> .


Question: Why are unknown coordinates denoted by characters found at the end of the alphabet?

Answer: This custom comes from a convention of algebra

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. <hl> This custom comes from a convention of algebra , which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities. <hl>

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. <hl> This custom comes from a convention of algebra <hl>, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

<hl> This custom comes from a convention of algebra <hl> , which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.


Question: What is denoted by letters near the beginning of the alphabet?

Answer: given quantities

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. <hl> This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities . <hl>

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for <hl> given quantities <hl>.

This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for <hl> given quantities <hl> .


Question: What does the z axis represent?

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left.  If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

Answer: height

<hl> For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). <hl> Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent <hl> height <hl> (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.


Question: What is the convention for the orientation of the x axis in three dimensional systems?

Answer: toward the viewer, biased either to the right or left

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). <hl> Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left . <hl> If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis <hl> toward the viewer, biased either to the right or left <hl>. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

Furthermore, there is a convention to orient the x-axis <hl> toward the viewer, biased either to the right or left <hl> .


Question: What determines the z axis' pointing down and to the left or down and to the right?

Answer: the presumed viewer or camera perspective

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. <hl> In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective . <hl> In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on <hl> the presumed viewer or camera perspective <hl>. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on <hl> the presumed viewer or camera perspective <hl> .


Question: If the x and y axis are shown vertically and horizontally, how should the z axis be shown in a diagram?

Answer: pointing "out of the page" towards the viewer or camera

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. <hl> If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera . <hl> In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown <hl> pointing "out of the page" towards the viewer or camera <hl>. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown <hl> pointing "out of the page" towards the viewer or camera <hl> .


Question: What do all laws of physics and math assume?

Answer: right-handedness

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. <hl> All laws of physics and math assume this right-handedness , which ensures consistency. <hl>

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this <hl> right-handedness <hl>, which ensures consistency.

All laws of physics and math assume this <hl> right-handedness <hl> , which ensures consistency.


Question: What is the reflection of the point (-x, y) across the second coordinate axis?

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror.  Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). In more generality, reflection across a line through the origin making an angle

θ

{\displaystyle \theta }

 with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

Answer: (the Y-axis)

<hl> If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis) , as if that line were a mirror. <hl> Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis <hl> (the Y-axis) <hl>, as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis <hl> (the Y-axis) <hl> , as if that line were a mirror.


Question: What is the first coordinate axis?

Answer: the X-axis

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror. <hl> Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis ( the X-axis ). <hl> In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (<hl> the X-axis <hl>). In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis ( <hl> the X-axis <hl> ).


Question: What is the second coordinate axis?

Answer: the Y-axis

<hl> If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis ( the Y-axis ), as if that line were a mirror. <hl> Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (<hl> the Y-axis <hl>), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis ( <hl> the Y-axis <hl> ), as if that line were a mirror.


Question: What is the reflection of the point (x, -y) across the first coordinate axis?

Answer: (the X-axis)

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror. <hl> Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis) . <hl> In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis <hl> (the X-axis) <hl>. In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis <hl> (the X-axis) <hl> .


Question: What describes an angle?

Answer: θ

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). <hl> In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where <hl>

If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X-axis). In more generality, reflection across a line through the origin making an angle <hl> θ <hl> {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where


Question: In standard orientation, in which way does the z axis point?

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line.  The two possible coordinate systems which result are called 'right-handed' and 'left-handed'.  The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

Answer: up

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. <hl> The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- <hl> and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points <hl> up <hl> (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.


Question: What determines the line along which the z axis lies?

Answer: the x- and y-axes

<hl> Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. <hl> The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

Once <hl> the x- and y-axes <hl> are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.


Question: In standard orientation, in what orientation is the xy-plane?

Answer: horizontal

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. <hl> The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive. <hl>

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is <hl> horizontal <hl> and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.


Question: How many possible directions does the z axis have?

Answer: two

<hl> Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. <hl> The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are <hl> two <hl> possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.


Question: What are the two possible coordinate systems determined by the z axis direction?

Answer: 'right-handed' and 'left-handed'

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. <hl> The two possible coordinate systems which result are called 'right-handed' and 'left-handed' . <hl> The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called <hl> 'right-handed' and 'left-handed' <hl>. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

The two possible coordinate systems which result are called <hl> 'right-handed' and 'left-handed' <hl> .


Question: What may create distortion in regard to coordinate systems?

Figure 7 depicts a left and a right-handed coordinate system.  Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer.  The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases).  Hence the red arrow passes in front of the z-axis.

Answer: a three-dimensional object is represented on the two-dimensional screen

Figure 7 depicts a left and a right-handed coordinate system. <hl> Because a three-dimensional object is represented on the two-dimensional screen , distortion and ambiguity result. <hl> The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.

Figure 7 depicts a left and a right-handed coordinate system. Because <hl> a three-dimensional object is represented on the two-dimensional screen <hl>, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.

Because <hl> a three-dimensional object is represented on the two-dimensional screen <hl> , distortion and ambiguity result.


Question: What axis meant to point away from the observer?

Answer: the "middle" axis

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) <hl> is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. <hl> The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas <hl> the "middle" axis <hl> is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.


Question: What axis is meant to point towards the observer?

Answer: The axis pointing downward

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. <hl> The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. <hl> The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. <hl> The axis pointing downward <hl> (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.


Question: In what manner is the red circle to the horizontal xy-plane?

Answer: parallel

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. <hl> The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). <hl> Hence the red arrow passes in front of the z-axis.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. The red circle is <hl> parallel <hl> to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.


Question: What indicates rotation from the x-axis to the y-axis?

Answer: The red circle

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. <hl> The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). <hl> Hence the red arrow passes in front of the z-axis.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle" axis is meant to point away from the observer. <hl> The red circle <hl> is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.


Question: What kind of coordinate system is figure 8?

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner".  This corresponds to the two possible orientations of the coordinate system.  Seeing the figure as convex gives a left-handed coordinate system.  Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Answer: right-handed

<hl> Figure 8 is another attempt at depicting a right-handed coordinate system. <hl> Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Figure 8 is another attempt at depicting a <hl> right-handed <hl> coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.


Question: What kind of coordinate system do you have if you see the figure as convex?

Answer: between a convex cube a

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. <hl> Many observers see Figure 8 as "flipping in and out" between a convex cube a nd a concave "corner". <hl> This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" <hl> between a convex cube a <hl>nd a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Many observers see Figure 8 as "flipping in and out" <hl> between a convex cube a <hl> nd a concave "corner".


Question: What do many observers see figure 8 as flipping in and out between?

Answer: convex cube and a concave "corner"

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. <hl> Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner" . <hl> This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a <hl> convex cube and a concave "corner" <hl>. This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Many observers see Figure 8 as "flipping in and out" between a <hl> convex cube and a concave "corner" <hl> .


Question: What is caused by projecting the three-dimensional coordinate system into the plane?

Answer: ambiguity

Figure 8 is another attempt at depicting a right-handed coordinate system. <hl> Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. <hl> Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an <hl> ambiguity <hl> caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.


Question: What is the correct way to view Figure 8

Answer: imagine the x-axis as pointing towards the observer

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. <hl> Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner. <hl>

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to <hl> imagine the x-axis as pointing towards the observer <hl> and thus seeing a concave corner.


Question: Unit vectors go in what axis's?

where

i

=

(

1

0

)

{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}}

, and

j

=

(

0

1

)

{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}}

 are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates

(
x
,
y
,
z
)

{\displaystyle (x,y,z)}

 can be written as:

Answer: x-axis and y-axis

<hl> where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). <hl> Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be written as:

where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the <hl> x-axis and y-axis <hl> respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be written as:


Question: Standard basis may also be called what?

Answer: versors

<hl> where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors ) <hl> . Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be written as:

where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as <hl> versors <hl>). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be written as:

where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as <hl> versors <hl> )


Question: The x-axis and y-axis are generally called what?

Answer: standard basis

<hl> where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). <hl> Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be written as:

where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the <hl> standard basis <hl> (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be written as:


Question: What are the 3 axis's in in a three-dimensional system?

Answer: x,y,z

where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). <hl> Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle ( x,y,z )} can be written as: <hl>

where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (<hl> x,y,z <hl>)} can be written as:

Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle ( <hl> x,y,z <hl> )} can be written as:


Question: Is there a way to multiply vector to obtain another vector?

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

Answer: There is no natural interpretation

<hl> There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. <hl> In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

<hl> There is no natural interpretation <hl> of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.


Question: How do you identify point coordinates in a two-dimensional plane?

Answer: z = x + iy

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. <hl> In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy . <hl> Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number <hl> z = x + iy <hl>. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number <hl> z = x + iy <hl> .


Question: i is identified by a point with what coordinates?

Answer: (0, 1)

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. <hl> Here, i is the imaginary unit and is identified with the point with coordinates (0, 1) , so it is not the unit vector in the direction of the x-axis. <hl> Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates <hl> (0, 1) <hl>, so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

Here, i is the imaginary unit and is identified with the point with coordinates <hl> (0, 1) <hl> , so it is not the unit vector in the direction of the x-axis.


Question: What does i stand for?

Answer: imaginary unit

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. <hl> Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. <hl> Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the <hl> imaginary unit <hl> and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.


Question: What kind of coordinates have many uses in the real world?

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world.  However, three constructive steps are involved in superimposing coordinates on a problem application.  1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates.  2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.

Answer: Cartesian

<hl> Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. <hl> However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.

<hl> Cartesian <hl> coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.


Question: How many steps are involved in entering coordinates on a problem application?

Answer: three

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. <hl> However, three constructive steps are involved in superimposing coordinates on a problem application. <hl> 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, <hl> three <hl> constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.


Question: Distance units are decided by defining what?

Answer: spatial size

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. <hl> 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. <hl> 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the <hl> spatial size <hl> represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.


Question: What is assigned to a specific location or landmark?

Answer: origin

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) <hl> An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes. <hl>

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An <hl> origin <hl> must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for (n-1) of the n axes.


Question: What units make sense to use when entering 3D coordinates all over the Earth?

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense?   Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin?  Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits).  Finally, how to orient X, Y and Z axis directions?   The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices.  In order to obey the right hand rule, the Y-axis must point out from the geocenter to  [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City?   Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ].  GPS navigation relies on such geocentric coordinates.

Answer: Kilometers

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? <hl> Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. <hl> Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? <hl> Kilometers <hl> are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.


Question: What is the distance between the Equator and the North Pole?

Answer: 10,000 km

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial... <hl> 10,000 km equalling the surface distance from the Equator to the North Pole. <hl> Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...<hl> 10,000 km <hl> equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.


Question: Where is a good location along the Equator to help define the X-axis?

Answer: Prime Meridian

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. <hl> A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. <hl> Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the <hl> Prime Meridian <hl> stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.


Question: What are the coordinates of the Empire State Building?

Answer: longitude = −73.985656, latitude = 40.748433

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? <hl> Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = <hl> 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ <hl> longitude = −73.985656, latitude = 40.748433 <hl> ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.


Question: What equation do you use to determine the radius of the Earth?

Answer: 40,000/2π

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? <hl> Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π , and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = <hl> [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.

Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X, Y and Z axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the Prime Meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = <hl> 40,000/2π <hl>, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.

Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = <hl> 40,000/2π <hl> , and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] =


Question: What kind of apps employ identical units along all axes?

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.).  Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done.  (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.)  Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.

Answer: spatial apps

<hl> While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). <hl> Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.

While <hl> spatial apps <hl> employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.


Question: How many variables can be determined using the algebra of Cartesian coordinates?

Answer: four or more variables

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). <hl> Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables , so that certain calculations involving many variables can be done. <hl> (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to <hl> four or more variables <hl>, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.

Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to <hl> four or more variables <hl> , so that certain calculations involving many variables can be done.


Question: What kinds of units of measurement can be associated with different axes?

Answer: kilograms, seconds, pounds, etc

<hl> While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc .). <hl> Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as <hl> kilograms, seconds, pounds, etc <hl>.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as <hl> kilograms, seconds, pounds, etc <hl> .).


Question: What is it helpful to use when trying to determine in two or three dimensional relationships?

Answer: geometry of Cartesian coordinates

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) <hl> Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables. <hl>

While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the <hl> geometry of Cartesian coordinates <hl> in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.


Question: What is used to to show the relation of all points satisfying that relation?

The graph of a function or relation is the set of all points satisfying that function or relation. For a function of one variable, f, the set of all points (x, y), where y = f(x) is the graph of the function f. For a function g of two variables, the set of all points (x, y, z), where z = g(x, y) is the graph of the function g. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.

Answer: graph of a function or relation

<hl> The graph of a function or relation is the set of all points satisfying that function or relation. <hl> For a function of one variable, f, the set of all points (x, y), where y = f(x) is the graph of the function f. For a function g of two variables, the set of all points (x, y, z), where z = g(x, y) is the graph of the function g. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.

The <hl> graph of a function or relation <hl> is the set of all points satisfying that function or relation. For a function of one variable, f, the set of all points (x, y), where y = f(x) is the graph of the function f. For a function g of two variables, the set of all points (x, y, z), where z = g(x, y) is the graph of the function g. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.