Question

A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?

Answer 1

Hint: A vector is a physical quantity that has both magnitude and direction. In addition to these two parameters, a vector follows the laws of vector addition. Two vectors cannot be added by using algebraic laws of addition. Any physical quantity that has both magnitude and direction and also follows the laws of vector addition is considered to be a vector.

Complete step by step answer:
For a physical quantity to be classified as a vector, it must have both magnitude and direction and in addition to that, must follow the laws of vector addition. Any quantity that fails to satisfy any of these conditions cannot be classified as a vector.
Though rotation of a body can be specified by the direction of the axis of rotation and the magnitude of the angle of rotation about the axis, it cannot be classified as a vector since it does not follow the laws of vector addition. In fact rotations are added algebraically similar to scalars. However, infinitesimal rotations about a certain small angle can be approximated to be vectors.
Therefore, any rotation cannot be considered as a vector.

Note:
Another such quantity that has both magnitude and direction but is not a vector is electric current. Electric current has a certain magnitude and direction of flow. However, two currents are always added algebraically similar to scalars. This can be observed in a junction, where two or more currents entering the junction combine algebraically to form a current exiting the junction.
Students must know of this concept and not get confused by thinking that any physical quantity that has a magnitude and direction is a vector.