Introduction to Vectors

To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a ship or a helicopter for a rescue mission, the rescue team needs to know not only the distance to the distress signal, but also the direction from which the signal is coming, so that they can get to it as quickly as possible. Physical quantities specified completely with a number of units (magnitude) and a direction are called vector quantities. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called vectors. We denote vector quantities by an italicized letter with an arrow above it. A quantity with only a magnitude and no direction is a scalar quantity.Therefore, weight is a vector quantity, whereas mass is a scalar quantity. Similarly, speed is a scalar quantity, whereas velocity is a vector quantity.

Suppose you tell a friend on a camping trip that you have discovered a terrific fishing hole 6 km from your tent. It is unlikely your friend would be able to find the hole easily unless you also provide the direction in which it can be found from your campsite. You may say, for example, "Walk about 6 km northeast from my tent."The key concept here is that you must give two pieces of information, namely the distance or magnitude (6 km) and the direction (northeast). A change in position, such as from the tent to the fishing hole in this example, is known as displacement. This is an example of a vector quantity. Geometrically, vectors are represented as arrows; their length (which is a positive number) gives the magnitude, indicated by placing the absolute value notation around the symbol that denotes the vector. The point of the arrow gives the direction, and the arrowhead marks the end point of the vector.

Two vectors are equal if and only if they have equal magnitudes and the same direction. Two vectors that have identical directions are said to be parallel vectors—meaning they are parallel to each other. However, if a vector points in the opposite direction, which is exactly 180° to the first vector, they are said to be antiparallel. Two vectors with directions perpendicular to each other are said to be orthogonal vectors.

This text is adapted from Openstax, University Physics Volume 1, Section 2.1: Scalars and Vectors.