Question

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

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

Answer 1

The correct option is B

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

With respect to the first coordinate system,
$\overrightarrow{a}=2p\hat{i}+\hat{j}$
On rotation, let $\overrightarrow{b}$ be the vector with components (p+1) and 1 so that,
$\overrightarrow{b}=(p+1)\hat{i}+\hat{j}$
Rotation of the coordinate system does not change the magnitude of the vector
$\Rightarrow ,|\overrightarrow{a}|=|\overrightarrow{b}|\Rightarrow a^2=b^2\Rightarrow 4p^2+1=(p+1{)}^2+1\Rightarrow 4p^2=(p+1{)}^2\Rightarrow 2p=\pm (p+1)\Rightarrow 3p=-1orp=1\therefore p=-\frac{1}{3}orp=1$