Question

A vector of length $l$ is turned through the angle $\theta$ about its tail. What is the change in the position vector of its head-

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

Answer 1

The correct option is B $2l\sin \left(\theta /2\right)$

Consider the vector be ${\overrightarrow{r}}_1$ of length $L$

On turning through angle $\theta$ be ${\overrightarrow{r}}_2$

$\begin{array}{c}\Delta \overrightarrow{r}={\overrightarrow{r}}_2-{\overrightarrow{r}}_1\\ =\sqrt{{r_2}^2+{r_1}^2-2r_2{r}_1,\cos \theta }\end{array}$

But $r_1=r_2=L$

$\begin{array}{c}=\sqrt{L^2+L^2-2L^2\cos \theta }\\ =\sqrt{2L^2\left(1-\cos \theta \right)}\\ =\sqrt{2L^2\times 2{\sin }^2\frac{\theta }{2}}\\ =2L\sin \frac{\theta }{2}\end{array}$

Hence, The Option $B$ is the correct answer $2L\sin \frac{\theta }{2}$