What is a vector? A vector, also called vector quantity, is a quantity that has both magnitude and direction. For example, suppose you are pulling a cart up a 25 degrees incline and it requires an effort of 75 pounds.
This situation is describing a vector because it has a magnitude and it has a direction. The magnitude is 75 pounds and the direction is 30 degrees from the horizontal.
Geometrically, a vector is a directed line segment with an initial point and a terminal point such as the one shown below.
We can use boldface letters such as v, w, s to denote vectors. Notice the boldface letter v used above to denote the vector shown in blue.
There are four possible relationships between two vectors v and w.
v = w because the vectors have the same magnitude and the same direction.
v ≠ w because the vectors have the same direction, but different magnitudes.
v ≠ w because the vectors have the same magnitudes, but different directions.
v ≠ w because the vectors have the same magnitudes, but opposite directions.