Question

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

• A. $-1$
• B. $\frac{-1}{3}$
• C. $1$
• D. $2$

Answer 1

Answer: (B), (C)

The correct options are
B $\frac{-1}{3}$
C $1$

We have, $\overrightarrow{a}=2p\hat{i}+\hat{j}$
On rotation, let $\overrightarrow{b}$ be the vector with components (p + 1) and 1 so that $\overrightarrow{b}=(p+1)\hat{i}+\hat{j}$.
Now, $|\overrightarrow{a}|=|\overrightarrow{b}|$ or $a^2=b^2$
$\Rightarrow 4p^2+1=(p+1{)}^2+1$
or $4p^2=(p+1{)}^2$
or $2p=\pm (p+1)$
$\Rightarrow 3p=-1$ or $p=1$
$\therefore p=-\frac{1}{3}$ or $p=1$