Vectors in the Plane and Three dimensional Space
What is a Vector?
Vectors are often used in physics to represent quantities like force and velocity when we need to specify both the magnitude and direction of those quantities. Usually, we represent a vector visually by an arrow. In the applet below, vectors in and are shown.
Remark: There is a very special vector that does not have any direction. What is it?
Usually, vectors are drawn as arrows pointing out from the origin (the point where all axes intersect). On a coordinate space (either 2D or 3D), the coordinates of the point at the arrowhead uniquely determine the vector. Therefore, for the vectors v in and u in , we can express them mathematically as follows:
and
where and are points at the arrowhead of v and u respectively.
(You can freely drag the points on the arrowheads of both vectors in the applet above to change the length and the direction of the vectors.)
Moreover, two vectors are regarded as equal if they have the same length and direction. Hence, sometimes we may shift the vector to another position if necessary, as long as the length and direction remain unchanged.
(You can freely drag the green vectors in the applet above to any position you like and they are considered as the same vectors as the black ones pointing out from the origin.)