Vectors

A vector is a quantity which possesses direction as well as magnitude. It is particularly useful in mechanics since force, velocity and acceleration are all vector quantities. By using the axis of rotation and the right-hand thread rule to denote direction, we can also regard torque (or moment of force), angular velocity and angular acceleration as vector quantities.

Bold type is used to indicate that a symbol a, say, represents a vector. It may be illustrated diagrammatically by an arrow, the length of which is proportional to the magnitude and the direction by the arrow. Changing the sign to −a just reverses the direction of the vector a, as in Figure 21.1.

The magnitude of the vector is the modulus ∣a∣ but written more simply as a in standard type. The symbol â is used to indicate a vector which has unit magnitude and the same direction as a. Hence, â is a unit vector and a = aâ.

If a and b are two vectors, their sum c = a + b is obtained by the triangle law of addition, as shown in Figure 21.2. By completing the parallelogram with the dotted lines in the diagram, we see how the triangle law corresponds to the parallelogram law in the parallelogram of forces. Correspondingly, c = a + b may be referred to as the vector sum or the resultant of a and b.