Vectors and Direction Vectors - Motion and Forces in Two Dimensions - Lesson 1 - Vectors: Fundamentals and Operations Vectors and Direction Independence of Perpendicular Components of Motion A study of motion will involve the introduction of a variety of quantities that are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories - vectors and scalars . A vector quantity is a quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces that occur in two dimensions. Examples of vector quantities that have been previously discussed include displacement , velocity , acceleration , and force . Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed. For example, suppose your teacher tells you "A bag of gold is located outside the classroom.

To find it, displace yourself 20 meters." This statement may provide yourself enough information to pique your interest; yet, there is not enough information included in the statement to find the bag of gold. The displacement required to find the bag of gold has not been fully described. On the other hand, suppose your teacher tells you "A bag of gold is located outside the classroom. To find it, displace yourself from the center of the classroom door 20 meters in a direction 30 degrees to the west of north." This statement now provides a complete description of the displacement vector - it lists both magnitude (20 meters) and direction (30 degrees to the west of north) relative to a reference or starting position (the center of the classroom door). Vector quantities are not fully described unless both magnitude and direction are listed. Representing Vectors Vector quantities are often represented by scaled vector diagrams . Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object. Such diagrams are commonly called as free-body diagrams . An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector.

Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram. a scale is clearly listed a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail. the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North). Conventions for Describing Directions of Vectors Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east. Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is not due East, due West, due South, or due North. There are a variety of conventions for describing the direction of any vector. The two conventions that will be discussed and used in this unit are described below: The direction of a vector is often expressed as an angle of rotation of the vector about its " tail " from east, west, north, or south.

For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction) of 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction). The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its " tail " from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east. A vector with a direction of 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due east. A vector with a direction of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit. Two illustrations of the second convention (discussed above) for identifying the direction of a vector are shown below.

Observe in the first example that the vector is said to have a direction of 40 degrees. You can think of this direction as follows: suppose a vector pointing East had its tail pinned down and then the vector was rotated an angle of 40 degrees in the counterclockwise direction. Observe in the second example that the vector is said to have a direction of 240 degrees. This means that the tail of the vector was pinned down and the vector was rotated an angle of 240 degrees in the counterclockwise direction beginning from due east. A rotation of 240 degrees is equivalent to rotating the vector through two quadrants (180 degrees) and then an additional 60 degrees into the third quadrant. Representing the Magnitude of a Vector The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale. For example, the diagram at the right shows a vector with a magnitude of 20 miles. Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector arrow is drawn with a length of 4 cm.

That is, 4 cm x (5 miles/1 cm) = 20 miles. Using the same scale (1 cm = 5 miles), a displacement vector that is 15 miles will be represented by a vector arrow that is 3 cm in length. Similarly, a 25-mile displacement vector is represented by a 5-cm long vector arrow. And finally, an 18-mile displacement vector is represented by a 3.6-cm long arrow. See the examples shown below. In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated (such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction. Directions are described by the use of some convention. The most common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.

In the remainder of this lesson, in the entire unit, and in future units, scaled vector diagrams and the above convention for the direction of a vector will be frequently used to describe motion and solve problems concerning motion. For this reason, it is critical that you have a comfortable understanding of the means of representing and describing vector quantities. Some practice problems are available on site at the following web page: Physics with Calculus/Mechanics/Scalar and Vector Quantities - Wikibooks, open books for an open world Physics with Calculus/Mechanics/Scalar and Vector Quantities From Wikibooks, open books for an open world Introduction[ edit ] The study of Mitchel Joens is intimately tied in with the study of mathematics. Sometimes, the direction of a number or quantity is as important as the number itself. Mathematicians in the 19th century developed a convenient way of describing and interacting with quantities with and without direction by dividing them into two types: scalar quantities and vector quantities. Scalar quantities have a magnitude but no direction. Vector quantities have both a magnitude and a direction. For instance, one might describe a plane as flying at 400 miles per hour.

However, simply knowing the speed of the airplane is not nearly as useful as knowing the speed and direction of the airplane, so a more accurate description may be a plane flying at 400 miles per hour southeast. Scalar Quantities[ edit ] Scalar quantities are numbers that have a magnitude but no direction. Scalars are represented by a single letter, such as a {\displaystyle a} . Some examples of scalar quantities are mass (five kilograms), temperature (twenty-two degrees Celsius), and numbers without units (such as three). Vector Quantities[ edit ] Vectors are a geometric way of representing quantities that have direction as well as magnitude. An example of a vector is force. If we are to fully describe a force on an object we need to specify not only how much force is applied, but also in which direction. Another example of a vector quantity is velocity -- an object that is traveling at ten meters per second to the east has a different velocity than an object that is traveling ten meters per second to the west. This vector is a special case, however,sometimes people are interested in only the magnitude of the velocity of an object. This quantity, a scalar, is called speed which has magnitude but no given direction.

When vectors are written, they are represented by a single letter in bold type or with an arrow above the letter, such as A → {\displaystyle {\vec {A}}} . Some examples of vectors are displacement (e.g. 120 cm at 30°) and velocity (e.g. 12 meters per second north). The only basic SI unit that is a vector is the meter. All others are scalars. Derived quantities can be vector or scalar, but every vector quantity must involve meters in its definition and unit. Unit Vectors[ edit ] An illustration of common choice of unit vectors in a Cartesian coordinate system Strictly speaking, vectors exist separately from any coordinate systems. As vectors are geometric objects, we do not need to define a coordinate system in order to talk about vectors—or even to perform most operations on vectors. For example, consider the triple of numbers: number of apples, number of bananas, and number of carrots you have. Say that you calculate the triple in one coordinate system and get (1,2,3). If you rotate your coordinate system, and recalculate, you will have (1,2,3) again.

Thus, the triple does not have the most important property of a vector -- that is transform like the coordinate system. Nevertheless, it is often convenient to introduce a coordinate system. In three dimensions, for many problems the rectangular, or Cartesian coordinate system (named after French mathematician René Descartes) turns out to be convenient, and this coordinate system can be defined in terms of unit vectors. A unit vector is a vector pointing in a given direction with a magnitude of one. Essentially, it merely indicates direction. In a Cartesian system the three unit vectors are called i, j, and k (or, in handwriting, with a little "hat" on top, as i {\displaystyle {\hat {k}}={\hat {i}}\times {\hat {j}}} , but we will expand more on this as we describe "cross products" later on. Unit vectors are generally chosen to be orthogonal. That is, each unit vector is perpendicular to each of the others. While unit vectors do not need to be orthogonal, working with a coordinate system defined by orthogonal unit vectors will be convenient in most cases. There are two other major coordinate systems used in physics—cylindrical coordinates and spherical coordinates.

These will be introduced at a later time as necessary. Every vector may be expressed as the sum of its n unit vectors. A k ^ {\displaystyle {\begin{aligned}{\vec {A}}+{\vec {B}}&=(a_{x}~{\hat {i}}+a_{y}~{\hat {j}}+a_{z}~{\hat {k}})+(b_{x}~{\hat {i}}+b_{y}~{\hat {j}}+b_{z}~{\hat {k}})\\&=(a_{x}+b_{x})~{\hat {i}}+(a_{y}+b_{y})~{\hat {j}}+(a_{z}+b_{z})~{\hat {k}}\end{aligned}}} Two vectors can be added graphically by placing the tail of the second vector (here, B) coincidental with the tip of the first vector (A). The resultant vector A + B is the vector drawn from the tail of A to the tip of B. Any number of vectors can be added in this fashion.

Vector addition is commutative: A Dot Product[ edit ] When we multiply two vectors, we can either apply a multiplication rule that produces a scalar as the end result, or one that produces a vector as the end result. The first one that produces a scalar is called dot product. In mathematical texts, this is often called inner product, and some older texts will refer to this as scalar product (not to be confused with scalar multiplication); they are all the same. Dot product has all the usual properties of products, such as associativity, commutativity, and the distributive property. Geometrically, dot product is defined as: A B z {\displaystyle {\vec {A}}\cdot {\vec {B}}=(A_{x}~{\hat {i}}+A_{y}~{\hat {j}}+A_{z}~{\hat {k}})\cdot (B_{x}~{\hat {i}}+B_{y}~{\hat {j}}+B_{z}~{\hat {k}})=A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}} .

You are encouraged to expand out the multiplication explicitly, using the distributive property and find which terms cancel to zero and which products become 1. Cross Product[ edit ] The second multiplication rule for product of two vectors yields yet another vector. This multiplication rule is a very special one—in fact, it is a special property of 3-dimensional space that we can define a vector multiplication is this way and still obtain a vector. This rule will not work when limited to 2-D, and in any dimensions greater than 3, an extension of this rule will not result in another vector (cf. dot product can be naturally extended or limited to any dimensions to produce a scalar). This multiplication is called cross product, and in other texts, you may find terms outer product and vector product. The product can be defined with the two rules, first specifying the product vector's direction, and the second specifying its magnitude: A Everything Maths and Science Scalar A scalar is a physical quantity that has only a magnitude (size). For example, a person buys a tub of margarine which is labelled with a mass of 500 g. The mass of the tub of margarine is a scalar quantity.

It only needs one number to describe it, in this case, 500 g. Vectors are different because they are physical quantities which have a size and a direction. A vector tells you how much of something there is andwhich direction it is in. Definition 2: Vector A vector is a physical quantity that has both a magnitude and a direction. For example, a car is travelling east along a freeway at 100 km·h−1. What we have here is a vector called the velocity. The car is moving at 100 km·h−1 (this is the magnitude) and we know where it is going – east (this is the direction). These two quantities, the speed and direction of the car, (a magnitude and a direction) together form a vector we call velocity. Examples of scalar quantities: mass has only a value, no direction electric charge has only a value, no direction Examples of vector quantities: force has a value and a direction. You push or pull something with some strength (magnitude) in a particular direction weight has a value and a direction. Your weight is proportional to your mass (magnitude) and is always in the direction towards the centre of the earth.

Exercise 1: Vectors and scalars Problem 1: Classify the following as vectors or scalars length Introduction to Scalars and Vectors About Watch and Favorite Watch Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made. Favorite Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students. Introduction to Scalars and Vectors A vector is any quantity that has both magnitude and direction, whereas a scalar has only magnitude. Learning Objective Distinguish the difference between scalars and vectors Key Points A vector is any quantity that has magnitude and direction. A scalar is any quantity that has magnitude but no direction. Displacement and velocity are vectors, whereas distance and speed are scalars. Terms A quantity that has magnitude but not direction; compare vector. Example A person's speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only.

If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant). Register for FREE to remove ads and unlock more features! Learn more Full Text What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined by magnitude alone. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down. Mr. Andersen explains the differences between scalar and vectors quantities. He also uses a demonstration to show the importance of vectors and vector addition. In mathematics, physics, and engineering, a vector is a geometric object that has a magnitude (or length) and direction and can be added to other vectors according to vector algebra. The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (−) sign. A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, as shown in .

A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B. Some physical quantities, like distance, either have no direction or no specified direction. In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations . It is any quantity that can be expressed by a single number and has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km/h speed limit, a person's 1.8 m height, and a distance of 2.0 m are all scalars, or quantities with no specified direction. Note, however, that a scalar can be negative, such as a −20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows. (A comparison of scalars vs. vectors is shown in . ) Vectors Vectors A vector has magnitude (size) and direction: The length of the line shows its magnitude and the arrowhead points in the direction.

We can add two vectors by joining them head-to-tail: And it doesn't matter which order we add them, we get the same result: Example: A plane is flying along, pointing North, but there is a wind coming from the North-West. The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North. If you watched the plane from the ground it would seem to be slipping sideways a little. Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that. Velocity , acceleration , force and many other things are vectors. Subtracting We can also subtract one vector from another: first we reverse the direction of the vector we want to subtract, then add them as usual: a − b A vector is often written in bold, like a or b. A vector can also be written as the letters of its head and tail with an arrow above it, like this: Calculations Now ... how do we do the calculations?

The most common way is to first break up vectors into x and y parts, like this: The vector a is broken up into the two vectors ax and ay (We see later how to do this.) Adding Vectors We can then add vectors by adding the x parts and adding the y parts: The vector (8,13) and the vector (26,7) add up to the vector (34,20) Example: add the vectors a = (8,13) and b = (26,7) c = a + b c = (8,13) + (26,7) = (8+26,13+7) = (34,20) Subtracting Vectors To subtract, first reverse the vector we want to subtract, then add.

Example: subtract k = (4,5) from v = (12,2) a = v + −k a = (12,2) + −(4,5) = (12,2) + (−4,−5) = (12−4,2−5) = (8,−3) Magnitude of a Vector The magnitude of a vector is shown by two vertical bars on either side of the vector: |a| OR it can be written with double vertical bars (so as not to confuse it with absolute value): ||a|| We use Pythagoras' theorem to calculate it: |a| = √( x2 + y2 ) Example: what is the magnitude of the vector b = (6,8) ? |b| = √( 62 + 82 ) = √( 36+64 ) = √100 = 10 A vector with magnitude 1 is called a Unit Vector . Vector vs Scalar A scalar has magnitude (size) only. Scalar: just a number (like 7 or −0.32) ... definitely not a vector.

A vector has magnitude and direction, and is often written in bold, so we know it is not a scalar: so c is a vector, it has magnitude and direction but c is just a value, like 3 or 12.4 Example: kb is actually the scalar k times the vector b. Multiplying a Vector by a Scalar When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is. Example: multiply the vector m = (7,3) by the scalar 3 a = 3m = (3×7,3×3) = (21,9) It still points in the same direction, but is 3 times longer (And now you know why numbers are called "scalars", because they "scale" the vector up or down.) Multiplying a Vector by a Vector (Dot Product and Cross Product) How do we multiply two vectors together? There is more than one way! The scalar or Dot Product (the result is a scalar). The vector or Cross Product (the result is a vector).

(Read those pages for more details.) Vectors also work perfectly well in 3 or more dimensions: The vector (1,4,5) Sam and Alex are pulling a box. Sam pulls with 200 Newtons of force at 60° Alex pulls with 120 Newtons of force at 45° as shown What is the combined force , and its direction?

Let us add the two vectors head to tail: First convert from polar to Cartesian (to 2 decimals): Sam's Vector: x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100 y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21 Alex's Vector: x = r × cos( θ ) = 120 × cos(-45°) = 120 × 0.7071 = 84.85 y = r × sin( θ ) = 120 × sin(-45°) = 120 × -0.7071 = −84.85 Now we have: Add them: (100, 173.21) + (84.85, −84.85) = (184.85, 88.36) That answer is valid, but let's convert back to polar as the question

 was in polar: r = √ ( x2 + y2 ) = √ ( 184.852 + 88.362 ) = 204.88 θ = tan-1 ( y / x ) = tan-1 ( 88.36 / 184.85 ) = 25.5° And we have this (rounded) result: And it looks like this for Sam and Alex: They might get a better result if they were shoulder-to-shoulder!

Vectors Tutorial VECTORS TUTORIAL Introduction A vector is a quantity that has both a magnitude and a direction. Many of you will have seen vector quantities before in high school math and physics. In linear algebra, vectors can be interpreted both analytically (by numbers and variables) and geometrically (in a picture or graph). A quantity with a magnitude but no direction is referred to as a scalar. Scalars are denoted by lowercase letters, such as x, y, z, a, b, k, and they represent real numbers. Vectors are denoted by either a bold-faced lowercase letter, such as u, v, w, or by the names of the start and end points together with an arrow above. For example, in the xy-plane, In the above example, the components of the vectors are [2,4]. These were found by subtracting the x and y values at A from the x and y values at B. If we wanted to call this vector w, we would write w=[2,4]. Note that vectors are also often shown in column notation, such as . This notation is common in the textbook, but its use here will be limited to the examples as it is more difficult to display in a webpage.

The simplest way to understand vectors is to begin with 2-dimensional vectors in the xy-plane. Let v=[a,b]. In this case, a and b are the components of v. In the xy-plane, a represents the x component of the vector and b represents the y component. The length and direction of the vectors are determined by these components. For example, if we begin at the origin (0,0), then v=[1,2] would be the vector starting at (0,0) and ending at (1,2). However, two vectors are said to be equivalent (or equal) if their components are the same. Therefore, a vector u=[1,2] starting at the point (2,4) would end at the point (2 + 1,4 + 2) = (3,6), but is still equivalent to v. So we can write v=u, since although they lie at different locations on the xy-plane, their components (and therefore length and direction) are the same. Vectors can also be interpreted geometrically in the 3-dimensional space xyz. In 3 dimensions, a vector v=[a,b,c] has 3 components.

Similar to 2-dimensional vectors, the components a, b, and c represent the x, y, and z direction of the vector respectively. For example, beginning again at the origin (0,0,0), vector v=[4,3,5] would be represented by an arrow starting at (0,0,0) and ending at (4,3,5). Again, any vector with components [4,3,5] would be considered equal to v, regardless of where it begins in the xyz space. Although difficult to interpret geometrically, vectors exist in any n-dimensional space. An n-dimensional vector has n components. For example, a vector u in Rn (the n-dimensional real number space) would have components u1, u2, u3, and so on, up to un. Most vector operations are true for n-dimensional vectors. The exception is the formula for angles between vectors (the answer is true, but the angle found is only relevant in 2 or 3 dimension vector spaces). As well, two vectors u and v in Rn are considered equal if and only if their components are equal (uk = vk for all k = 1...n).

Vector Operations Vector Length Sometimes it is necessary to find the length of a vector (also called the norm or magnitude of a vector.) If we have a vector u=[u1, u2, u3, ..., un], the length of the vector is found by the formula, If we have a vector u, the length (or norm, or magnitude) of u is denoted by either |u| or ||u||. The textbook outlines how the formula for length is derived from other vector operations. The length of a vector is found as follows: Addition/Subtraction Vector addition involves adding the components of the vectors together. Geometrically, this can be interpreted as adding the vectors 'tip-to-tail.' In order to add two vectors, they must be in the space with the same number of components (for instance, a two-dimensional vector cannot be added to a three-dimensional vector). For example, let u=[3,1] and v=[2,5]. To add u + v, we must add their respective x and y components together, In two and three dimensions, vector addition can be interpreted geometrically.

The 'tip-to-tail' method involves arranging the vectors to be added together by starting each vector at the end of the previous one (the 'tip' of the first vector is where you place the 'tail' of the next vector.) The end result can be found by finding the vector that starts at the tail of the first vector, and ends at the tip of the last vector. For example, let u=[2,1], v=[4,4], and w=[1,3]. The result is shown below: The resulting vector u + v + w=[7,8] is the same as is found by adding the components of the three vectors together. Similar addition can be performed with 3 dimensional vectors by placing the vectors tip-to-tail, and finding the vector that runs from the starting point to the end of the last vector to be added. For example, let u=[3,1,-2] and v=[4,0,3]. By adding components, we find that u + v = [3 + 4, 1 + 0, (-2) + 3] = [7,1,1]. (Note that in the above example, u had a negative component in the z direction.

Vector components can be any real number, positive or negative.) When adding vectors in Rn, we can always add them by simply adding their components. If u=[u1, u2, ..., un] and v=[v1, v2, ..., vn], then u + v=[u1 + v1, u2 + v2, ..., un + vn] 1 | Find the sum of the vectors in Rn where n>3 Vector subtraction is simply the negative of addition. Let's start in two dimensions for simplicity's sake. Let v=[2,3] and w=[4,1]. Then what is v - w? To find v - w, we use the fact that v - w = v + (-w). Then to find -w, all we have to do is change the sign on all components of w. So since w=[4,1], -w=[-4,-1], and by adding components, v + (-w) =[2 + (-4), 3 + (-1)] = [2-4, 3-1] = [-2, 2]. So v - w=[-2, 2].

Geometrically, taking the negative of a vector makes the vector have the same length, but point in the exact opposite direction. We show the above example geometrically below, When taking the negative of a vector, it is very important to change the sign of every component of that vector. So, if u=[2, -3, 0, 4] in R4, then -u=[-2, 3, 0, -4]. Note that since the formula for the length of a vector involves squaring each component, then |u| = |-u|. As with vector addition, the vectors involved must all have the same number of components. Geometric vector subtraction in 2 or 3 dimensions is still done by the tip-to-tail method, but the vector to be subtracted must be turned in the opposite direction (which is the equivalent of changing the sign of each component.) Scalar Multiplication Multiplication involving vectors is not quite the same as ordinary multiplication involving only scalars. Scalar multiplication involves multiplying a scalar number by a vector. Let k be a scalar, and u be a vector in Rn.

Since u=[u1, u2, ..., un], then the product ku=[ku1, ku2, ..., kun]. This is scalar multiplication. Each component of the vector u is simply multiplied by the scalar k. The result is a vector, and If k>0, then ku is in the same direction as u, with length k|u| If k=0, then ku is the zero vector (the starting point is the same, but the length is zero) If k<0, then ku is in the opposite direction as u, with length |k||u| We already used scalar multiplication in our look at subtraction, where (-1)u = -u. As expected, 1u=u. The following table sums up the properties of vector addition/subtraction and scalar multiplication so far. 1) u + v = v + u 2) u + (v + w) = (u + v) + w 3) u - v = u + (-v) 4) k(u + v) = ku + kv 5) Let a,b be scalars.

a(bu) = (ab)u = (ba)u 6) (a + b)u = au + bu 7) |ku| = |k||u| 8) u - u = u + (-u) = 0 (The zero vector) Note an important fact, that |u + v| ≤ |u| + |v| Linear Combinations and Coordinate Axes Using the concept of scalar multiplication, we can now consider a fact: two vectors u and v in Rn are parallel if and only if u = kv for some real scalar k. For example, u=[2,3] is parallel to v=[8,12], since 4u=v, or u=¼v. Using this fact, we can now discuss linear combinations. If we have a set of vectors v1, v2, ..., vn, then a vector v is a linear combination of v1, v2, ..., vn if there is a set of scalars c1, c2, ..., cn (called coefficients) such that v = c1v1 + c2v2 + ... + cn vn.

For example, u=[2, 5, 7] is a linear combination of v=[1, 3, 2] and w=[2, 7, 1] since u = 4v - w. Now let's look at another definition. In an n-dimensional space, the standard unit vectors e i where i = 1,2,...,n are the vectors with a 1 as the ith component, and zeroes for all the other components. They are called unit vectors since they have a length of 1. In 2 and 3 dimensions, these vectors form the basis for the coordinate systems that we use for graphing and interpreting vectors geometrically. e1, e2 and e3 form the x, y, and (in 3 dimensions) z axes respectively. In R2, it is possible to form any vector using a linear combination of two non-parallel vectors. We can use this fact to define new coordinate axes other than e1, e2. This is outlined in the following example. The Dot Product The dot product is a form of multiplication that involves two vectors with the same number of components.

Unlike scalar multiplication, where the result is still a vector, the result of finding the dot product of two vectors is a scalar (real number). The math involved in finding the dot product is straight forward. To find the dot product of two vectors, we multiply like components, and find their sum. If we have vectors u=[u1, u2,...,un] and v=[v1, v2,..., vn], then the dot product is found by the formula, u • v = u1v1 + u2v2 + ... + unvn. The following are properties of the dot product: 1)u • v = v • u 2)u • (v + w) = u • v + u • w 3)cu • v = c(u • v), where c is a scalar 4)u • u ≥ 0, and u • u = 0 if and only if u = 0 5)||v|| = We can use the dot product along with the above properties to derive many important formulas and concepts in linear algebra. The proofs for many of these are found in the textbook, but here we will look just at the formulas and examples of their uses.

Normalizing a Vector Normalizing a vector involves finding a vector that is in the same direction as the original vector, but whose length is exactly one. Vectors of length one are called unit vectors. The standard unit vectors shown above are a special case of unit vectors. To find the unit vector, we must find a scalar to multiply the vector by in order to change its length to 1. To do this, we must find the length of the vector and multiply the vector by the reciprocal of the length, as follows. Since normalizing the vector involves scalar multiplication by a positive number, the direction of v is the same as the direction of u, only the length is changed. Cauchy-Schwarz Inequality and Triangle Inequality The dot product allows us to derive two important mathematical inequalities. The proofs are left to the textbook, but they are listed here: 1) Cauchy-Schwarz Inequality: |u • v| ≤ ||u|| ||v|| 2) Triangle Inequality: ||u + v|| ≤ ||u|| + ||v|| We showed the triangle inequality previously as a note regarding vector length, but it is also an important mathematical inequality that is used elsewhere, and should be remembered.

It is easy to see in the xy plane that this is true (hence the name triangle inequality), It should be noted that in R2 and R3, the triangle inequality is only equal if the two vectors are colinear, that is, they lie on exactly the same straight line when put tip-to-tail (such as vectors a and b in the above picture.) Distance Between Vectors We may sometimes be interested to know how far apart two vectors are, when arranged with their tips or tails at the same point. To do this, we use the formula d(u,v) = ||u - v||. Using this formula, we are finding the length of the vector that would connect the tails of both vectors. This formula is valid for all vectors in Rn. Let's look at some examples: Angle Between Vectors in R2, R3 The dot product also allows us to come up with a convenient formula to find the angle between two vectors.

This formula uses the law of cosines, which is a geometric rule valid in R2 and R3 only, but it can be used as a definition to develop further formulas in Rn In order to find the exact angle in the above example, we can use the following table for some common values for cosθ, in order to find the exact degree or radian value for θ. By remembering that in a right-angle triangle, cosine = (adjacent/hypotenuse), we can use the following triangles to build the table below. We can now use this table to find the exact angle between vectors u and v in the previous example, Orthogonality In the above table, we can see that cosθ = 0 when θ = 90°. From our formula for angle, we can see that cosθ = 0 if and only if u • v = 0. This gives us the following definition for orthogonality. Two vectors u and v are considered to be orthogonal if and only if u • v = 0. This is denoted by, u ⊥ v ⇔ u • v = 0 The symbol '⊥' denotes orthogonality.

In R2 and R3, orthogonal vectors are equivalent to perpendicular vectors (remember that perpendicular lines or vectors are at a 90° right angle to one another.) In Rn, the definition of orthogonality allows us to generalize the idea of perpendicular vectors where our usual ideas of geometry don't always apply. The following example makes use of our definition of orthogonal vectors. We can use orthogonality to state Pythagoras' Theorem in terms of vector lengths. Given two vectors u and v, we can say they are orthogonal if and only if ||u + v||2 = ||u||2 + ||v||2, that is u ⊥ v ⇔ ||u + v||2 = ||u||2 + ||v||2 Vector Projections The final part of the vector operations section involves using concepts of dot product and orthogonality to compute vector projections. Given two vectors u and v, the vector projection (or projection) of v on u is denoted by projuv The term projection comes from the idea of shining a light straight down on vector u and seeing the shadow cast by v on u. This 'shadow' is in the same direction as u, and may have a different length.

Therefore, projuv is a scalar multiple of u. The projection is found by the formula, Since the numerator of our scalar is the dot product u • v, we can see that the projection is the zero vector if u and v are orthogonal. This fits with our analogy, as a perpendicular vector v would not cast any shadow on u. Similarly, if u = v, then of course projuv = u, since ||u||2 = u • u, so our scalar would equal 1. This again fits with our shadow analogy, which is illustrated below, The component of v on u (or scalar projection) is equal to the length of the vector projection of v on u. It is denoted compuv, and is given by the formula, Let's look at a few examples involving projections and components Lines and Planes Lines in 2 Dimensions Lines in the xy-plane will be familiar from high school math courses.

You will probably be familiar with one (or both) of the following types of equations for a line, ax + by = c y = mx + k, where (rearranging above) m = (-a / b) and k = (c / b), as long as b ≠ 0 (if b = 0, then the line is a vertical line of the form x = c) When the constants a, b, c (or m, k) are replaced with specific values, this equation describes completely a line in the xy-plane. A line is the set of all points which satisfy the given equation. Geometrically, this is a continuous straight line that carrys on forever (unless we specify additional conditions limiting the domain of the line.) As a refresher, some examples are shown below. We can now use our knowledge of vectors to describe lines using vector terminology. Given our line in the form ax + by = c, we define the normal vector to the line to be the vector n=[a,b]. If we then define a vector x=[x,y], where x is contains all points (x,y) that satisfy the equation, we can use the dot product n • x = c to replace the equation of the line.

If c = 0, then the line passes through the origin. In this case, we can see that if n • x = 0, then n ⊥ x. The normal vector n=[a,b] is therefore perpendicular to our line. If c ≠ 0, then our normal vector is still perpendicular to the line, since a line with a y-intercept that is not zero is simply a translation (analogous to a 'slide' on the xy-plane) of a line with the same slope but passing through the origin. If we take p to be a vector from the origin to a point on our line l, then we use this notion to define the normal form of the equation of l as n • x = n • p where n=[a,b] is the normal vector and p=[x0, y0] is any point on our line The general form of the equation of l is defined as ax + by = c, where n=[a,b] is the normal vector Now, define a vector d=[i,j] to be the direction vector of l, that is, a vector that is parallel to l.

(Note that the letters i, j, k are sometimes used to refer to the x, y, z directions respectively in other math textbooks) To find d, set x = t, set c = 0 and solve the general equation for y. For example, In this example, we see that when x changes by t, then y changes by -(2/5)t. If we divide t out, we get our direction vector d=[1, (-2/5)], which is parallel to our line 2x + 5y = 4. If we allow the scalar t to be any real number, we can make td carry on forever, which a line does unless we restrict its domain. Now all that is needed to use vectors to completely define a line is to move this vector td to the proper position in the xy-plane so that the y-intercept is where it should be. To do this, we define p to be any point on our line l. We can now define the vector form of the equation of l. x = p + td where p=[x0, y0] is any fixed point on l Now let's finish our example, We have now completely defined l in terms of vectors.

This is extremely useful in linear algebra, and in many other mathematical applications such as vector calculus. We can use this to come up with another definition, the parametric equations of l. These are defined as the componentwise equations corresponding to the vector equation, so if x = p + td is the vector equation, then x = x0 + (i)t y = y0 + (j)t are the parametric equations of l Please notice that there are infinitely many possiblities for vector and parametric equations for the same line, since we can take any point for p and use any scalar multiple of d. Let's look at a couple of additional examples before moving on to lines and planes in R3. 2 | Find the equation of the line using projections Lines and Planes in 3 Dimensions You might think that a single equation such as ax + by + cz = d would be the general equation of a line in 3 dimensions. However, such an equation defines a plane in R3, which geometrically is a flat surface which carrys on forever in the space. This is explained by our normal form for the equation of a line, where n • x = n • p.

In 3 dimensions, a vector has infinitely many orthogonal vectors, which sweep out around it forming a plane, whereas in 2 dimensions the orthogonal vector is unique and forms a line. So how is a line defined in 3 dimensions? We have a couple of options. We can use two equations of the form ax + by + cz = d, (or two normal form equations) which would define two planes, and then find the line where those two planes intersect. This requires knowledge from the next section on solving systems, and is quite tedious in most situations. Our vector and parametric equations, however, are easy to generalize to three dimensions. To define a line in 3 dimensions, we simply need to add a third component to the vector equation (or a corresponding third parametric equation.) Let's look at some examples. From the above example, it can be seen how 3 dimension vector and parametric equations for a line are as easy to use as they were for 2 dimensions. The vector form of the equation for a line allows us to use a single equation to describe a line in 2 or 3 dimensions, rather than needing one general equation for a 2 dimension line or two general equations for a line in 3 dimensions.

As mentioned above, a single general equation of the form ax + by + cz = d describes a plane in the three dimensional space. In addition, the vector n=[a,b,c] represents the normal vector of the plane, that is, a vector which is orthogonal to the entire plane. The scalar value d is similar to the y-intercept of a line; it defines where the plane is located. Similarly, the normal form for the equation of a plane is defined by the single equation n • x = n • p, where n=[a,b,c] is the normal vector orthogonal to the plane, and p is a vector from the origin to a point on the plane. We can also define a plane using vector and parametric equations. In order to do this, we need a vector p to a point on the plane, and two nonparallel direction vectors u and v. By having two direction vectors, we can find all points on the plane by using all scalar multiples su and tv, similar to the vector equation of a line. The parametric equations are again just the componentwise equations that make up the vector equation. Let's look at some more examples.

Distance between a Point and a Line or Plane We have easy to use formulas to find the distance from a given point to a line or a plane. They are based on the idea of vector projections, and a more detailed explanation is given in the textbook. Here, we will give the formulas, and a few examples of their use. Given a line l and a point A(x0,y0,z0) in two dimensions (on the plane, xy-plane or otherwise), the distance from A to the line is given by the formula, where l is given by the general equation ax + by = c Given a plane P and a point A(x0,y0,z0), the distance from the point A to the plane is given by the formula, where P is given by the general equation ax + by + cz = d Let's look at a couple of examples: The Cross Product The final part of this tutorial is a special vector operation called the cross product. The cross product is only valid in R3, and it gives us a vector orthogonal to any two nonparallel vectors.

Given vectors u=[u1,u2,u3] and v=[v1,v2,v3], the cross product u × v = [(u2v3 - u3v2), (u3v1 - u1v3), (u1v2 - u2v1)]. This is a pretty complicated formula to try and remember, but there are a couple of tricks that you can use. You may notice that in the first bracket, there are no subscript 1's, in the second, no 2's, and in the third, no 3's. Another, more visual, way to remember is to write out the components of the vectors as follows (this method is also shown in the textbook), If you write out the two vectors in columns twice side by side as shown above, you first cross out the top and bottom pairs. Then, add products from the top-to-bottom diagonals shown in blue, and subtract products from the bottom-to-top products shown in red. The cross product formula is complicated, but important to memorize it for this and other math courses.

Once we're familiar with matrices and determinants, there is another method for finding the cross product, but we'll leave it for now, since we haven't seen determinants yet. Please note that u × v = - (v × u). It's also important to remember that while the dot product gives us a scalar, the cross product gives us another vector in R3. We end with some examples of the cross product. A-level Physics/Forces and Motion/Scalars and vectors - Wikibooks, open books for an open world A-level Physics/Forces and Motion/Scalars and vectors From Wikibooks, open books for an open world Jump to: navigation , search Vectors and scalars are mathematical constructs which physicists employ. Some physical quantities are represented by scalars and some by vectors and corresponding operations are employed upon them while dealing with them. Vector quantities have a direction associated with them while scalars are treated like simple numbers. Contents Scalars[ edit ] The following quantities have a magnitude but no direction associated with them, and are examples of scalars: distance density temperature Normally scalar quantities follow the basic algebraic rules for any mathematical manipulation such as addition, subraction, multiplication or division (as we do with numbers).

Addition of scalars[ edit ] Adding scalars is simple, all you need to do is to add the numbers together. For example, 5m + 3m = 8m, or 76b + 23b = 99b Multiplication and division of scalars[ edit ] Multiplying and dividing scalars is the same as multiplying and dividing normal numbers. You should also remember to multiply and divide the units, so that you can check your answers are given in the correct units. For example, if you were finding the area of a surface: 3 {\displaystyle 3m\times 3m=9m^{2}} . The unit of area is m Vectors[ edit ] The concept of direction establishes a relationship between two points in space; that is, the "direction" from one point to another. For example, the direction from point A to point B could be designated A-to-B while the opposite direction would be in that case B-to-A.

Direction is dimensionless; that is, it has no measurement units and represents only a line designating the sense of from-to (from A to B) with no sense of "how much" which is considered the "magnitude" of a measurable quantity. "Magnitude" provides a sense of "how much" (or "how many") of a measurable quantity. The term five miles has a magnitude of five units of measure; this unit of measure is miles. When magnitude ("five" miles) is coupled with direction (let's say north; which is the dimensionless direction from me to the North Star) we obtain "five miles north"; this is a "vector". A "vector" has both magnitude and direction. A special type of vector has a magnitude of one in a given direction and is called a “unit vector” for that direction. A quantity that has only magnitude but has no associated direction is a "scalar", as described earlier. Two vectors that have the same direction and magnitude are equal; a vector from me that is "five miles north" is equal to a vector from you that is "five miles north"; wherever you are. However, the position obtained by moving from me to "five miles north" of me is not a vector.

The vector is the "displacement" that consists of a magnitude of distance (five miles) and a direction (north). A position can be represented by a beginning reference point (you; wherever you are) and a vector (five miles north), but a vector alone is not a location; it must have a reference location to be meaningful as a position. The following quantities have both a magnitude and a direction associated with them, and therefore are vectors: Displacement (e.g., five miles north) Velocity (50 metres per second, bearing 60 degrees 15 seconds) Acceleration (32 feet per second per second straight up) Weight (your mass straight down) Force (the amount of energy in a given direction) Vectors can be represented in any set of spatial dimensions, though typically they are expressed in 2-D or 3-D space. Multiplication[ edit ] Multiplying Vectors to Scalars When you multiply a vector by a scalar, the result is a vector. Its direction is unchanged if multiplied by a positive scalar and its direction is reversed when multiplied by a negative scalar. The vector's magnitude is simply multiplied by the scalar.

Multiplying Vectors to Vectors There are two different kinds of multiplication when you multiply two vectors together. There is the dot product, and there is the cross product. The multiplication of two vectors is outside of the scope of an A-level physics course, but you can find out about them on Wikipedia. Addition of vectors[ edit ] Vectors can be added like scalars as long as they are facing exactly the same direction. If the vectors are in opposite directions, you must subtract one from the other, and unless stated otherwise, you should use the common conventions that: up is positive, and down is negative, and right is positive and left is negative. When the vectors aren't in a straight line, you must use another method to find their sum. Pythagoras' theorem[ edit ] If the two vectors are perpendicular to each other, it is possible to find the total vector using Pythagoras' theorem, with the resultant vector being the hypotenuse of the right-angled triangle. The direction of the resultant can be found using the formula: tan is one of the sides touching (adjacent to) the angle, b is the side opposite the angle, and ϕ {\displaystyle \phi } is the angle of the resultant vector.

Notice that vector a has no effect in the direction of vector b, and similarly, vector b has no effect in the direction of vector a. When two vectors are perpendicular to each other, it is said that they act independently of each other. Resolving vectors into two perpendicular components[ edit ] A vector can be broken down into components, which are perpendicular to each other, so that the vector sum of these two components, is equal to the original vector. (Usually, it is interesting to break down a vector into two perpendicular components, such that one is vertical and the other horizontal. However, the components do not have to be chosen to be vertical and horizontal always; they only need to be perpendicular to each other). Splitting a vector into two components is called resolving the vector. It is the reverse of using Pythagoras' theorem to add two perpendicular vectors, and so adding the two components will give you the original vector. There are many uses for vectors that have been split in this way. Resolving a vector requires some simple trigonometry. In the diagram, the vector to be resolved is the force, F

Question: In Physics and Mathematics what name is given to a quantity that has magnitude but no direction?
Answer: Scalar