# Question #cec47

Apr 8, 2016

See explanation field below.

#### Explanation:

Let us first understand what is Angular Displacement

When a body is rotated about a specified axis, the angle of rotation of the radius vector or the distance moved by the body along the curved path is defined as its angular displacement.

The length of the arc of curved path is measured as an angle.

A point $P$ located on a rigid object rotates about a fixed axis through origin of coordinate system and perpendicular to the plane of the figure. (The axis of rotation is the $z$ axis.).
At time ${t}_{i}$ the radius vector of this point makes an angle ${\theta}_{i}$ with $x$ axis. Let the radius vector of this point be located at $Q$ after time ${t}_{f}$ due to rotation where it makes an angle ${\theta}_{f}$ with $x$ axis.

The angle moved by the radius vector is ${\theta}_{f} - {\theta}_{i}$. This is angular displacement of point $P$

Also, $\text{Arc length} P Q = r \left({\theta}_{f} - {\theta}_{i}\right)$

Even though angular displacement is an entity with a direction and a magnitude it does not obey the commutative law for addition; for finite rotations. Therefore, it is not a vector.

For example addition and multiplication are commutative, but subtraction and division are not. Similarly, angular displacement is non-commutative.
$A + B = B + A$ or $A \times B = B \times A$, commutative

$A - B \ne B - A$ or $A \div B \ne B \div A$, non-commutative.