Question

A vector $\overrightarrow{a}$ has components $2p$ and $1$ with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If $\overrightarrow{a}$ has components $p+1$ and $1$ with respect to the new system, then

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

Answer 1

The correct option is B $p=1$ or $-\frac{1}{3}$

$|\overrightarrow{a}|=|{\overrightarrow{a}}^{\prime }|$

$|2p\hat{i}+\hat{j}|=|(p+1)\hat{i}+\hat{j}|$

$\sqrt{4p^2+1}=\sqrt{(p+1{)}^2+1}$

$4p^2+1=(p+1{)}^2+1$

$2p=\pm (p+1)$

$2p=p+1$

$p=1$

$2p=-p-1$

$3p=-1$

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