Question

A vector of magnitude $a$ is rotated through angle $\theta$. What is the magnitude of the change in the vector?

• A. $2a\sin \theta$
• B. $2a\cos \theta$
• C. $2a\sin \left(\theta /2\right)$
• D. $2a\cos \left(\theta /2\right)$

Answer 1

The correct option is C $2a\sin \left(\theta /2\right)$

On rotation the magnitude of original vector does not change only its direction changes.
Let $\overrightarrow{a_1},\overrightarrow{a_2}$ denotes the initial and final direction and $\overrightarrow{c}$ denotes change in the vector.
so $\overrightarrow{c}=\overrightarrow{a_2}-\overrightarrow{a_1}\left|\overrightarrow{c}\right|=\sqrt{a_2^2+a_1^2-2a_1{a}_2\cos \theta }=\sqrt{a^2+a^2-2a^2\cos \theta }=2a\sin \left(\frac{\theta }{2}\right)$