Unizor - Vectors as Geometrical Objects
Channel: Zor Shekhtman on UNIZOR Education
Date: 2013-12-11

Consider a point moving on a plane or in a three-dimensional space along some trajectory. Our task is to geometrically represent the vector of its velocity, that is the speed and direction of a movement. The direction of its movement can be represented by an arrow originated at the location of a point or the beginning of coordinates and directed parallel to the movement (all parallel arrows point to the same direction). The speed of its movement can be reflected in the length of this arrow. So, this arrow with a specific length and direction is a graphical implementation of a vector.

What's important here is to notice that this geometrical representation of a vector is tightly related to a space where the vector operates. In a case of a two-dimensional movement on a plane, arrow on a plane represent the vectors. In three-dimensional case of movement the arrows in three-dimensional space are used.

In short, the dimensionality of a space where the vectors are considered determines what kind of arrows represent a vector. One-dimensional spaces are lines and curves. Two-dimensional spaces are planes and surfaces. Three-dimensional space is the space we used to think as the one we live in. Velocity of a movement along some trajectory is a vector represented by an arrow tangential to this trajectory. It's hard to imagine spaces with more than three dimensions and the geometrical representation of vectors is limited to no more than three dimensions.

Another example of using vectors is to represent physical forces. The force (mechanic, electromagnetic, gravitational etc.) also has the magnitude and the direction.