Question

A vector of magnitude $|\Psi |$ is turned through angle $\frac{\varphi }{2}$. The magnitude of change in the vector is given by

• A. $2|\Psi ||\cos \varphi /4|$
• B. $|\Psi ||\sin \varphi /4|$
• C. $|\Psi {|}^2|\sin \varphi /4{|}^2$
• D. $2|\Psi ||\sin \varphi /4|$

Answer 1

The correct option is D $2|\Psi ||\sin \varphi /4|$

By vector addition law, $\overrightarrow{OP}+\overrightarrow{PQ}=\overrightarrow{OQ}$

or $\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{-OP}$

The angle between vectors OQ and (-OP) is $(180-\varphi /2)$

Thus, $|\overrightarrow{PQ}{|}^2=r^2+r^2+2r^{2}cos(180-\varphi /2)=2r^2[1-cos(\varphi /2\left)\right]=2r^2\left(2si{n}^2\varphi /4\right)$

or $|\overrightarrow{PQ}|=2rsin\varphi /4=2|\Psi |sin\varphi /4$