Question

A vector has components $2p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated clockwise. If this respect to the new system has components $p+1$ and $1$, then?

• A. $p=0$
• B. $p=1\mathrm{or}p=-\frac{1}{3}$
• C. $p=-1$
• D. $p=1\mathrm{or}p\mathit{=}-1$

Answer 1

The correct option is B

$p=1\mathrm{or}p=-\frac{1}{3}$

Explanation for correct option:

Step-1: Express the given data.

Given vector in initial position having component components $2P$ and $1$is ${\overrightarrow{a}}_{initial}=\left(2p\right)\hat{i}+\hat{j}$

and after rotating vector become having component components $P+1$ and $1$ is ${\overrightarrow{a}}_{final}=\left(p+1\right)\hat{i}+\hat{j}$

Step-2: Equating magnitude.

$\Rightarrow$ $\left|{\overrightarrow{a}}_{initial}\right|=\left|{\overrightarrow{a}}_{final}\right|$

$\Rightarrow$ ${\left(2P\right)}^2+1={\left(P+1\right)}^2+{\left(1\right)}^2$

$\Rightarrow$ $4P^2+1=P^2+1+2P+1$

$\Rightarrow$ $3P^2-2P-1=0$

$\Rightarrow$ $3P^2-3P+P-5=0$

$\Rightarrow$ $3P\left(P-1\right)+1\left(P-1\right)=0$

$\Rightarrow$ $P=-\frac{1}{3},1$

Hence, option $\left(B\right)$ is correct.