Question

A vector a has components $3P$ and $1$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counterclockwise sense. If with respect to the new system, a has components $P+1$ and $\sqrt{10}$, then a value of $P$ is equal to:

• A. $1$
• B. $-1$
• C. $\frac{4}{5}$
• D. $-\frac{5}{4}$

Answer 1

The correct option is B

$-1$

Explanation for correct answer:

Step-1: Express the given data.

Given vector in initial position having component components $3P$ and $1$is ${\overrightarrow{a}}_{initial}=\left(3p\right)\hat{i}+\hat{j}$

and after rotating vector become having component components $P+1$ and $\sqrt{10}$ is ${\overrightarrow{a}}_{final}=\left(p+1\right)\hat{i}+\left(\sqrt{10}\right)\hat{j}$

Step-2: Equating magnitude.

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

$\Rightarrow$ ${\left(3P\right)}^2+1={\left(P+1\right)}^2+{\left(\sqrt{10}\right)}^2$

$\Rightarrow$ $9P^2+1=P^2+1+2P+10$

$\Rightarrow$ $8P^2-2P-10=0$

$\Rightarrow$ $4P^2-P-5=0$

$\Rightarrow$ $4P^2-5P+4P-5=0$

$\Rightarrow$$P\left(4P-5\right)+1\left(4P-5\right)=0$

$\Rightarrow$ $P=\frac{5}{4},-1$

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