Question

A vector $\overrightarrow{a}$ has component $2p$ and $1$ with respect to a rectangle Cartesian system. This system is rotated through a certain angle about the origin in the counterclockwise sense. If with respect to the new system, $\overrightarrow{a}$ has component $p+1$ and $1$, then :

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

Answer 1

Answer: (B)

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

We have $\overrightarrow{a}=2p\hat{i}+\hat{j}$,

After rotation we have $\overrightarrow{b}=(p+1)\hat{i}+\hat{j}$

Since rotation is made about the origin, $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|$

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

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