Question

A vector $a$ 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 counterclockwise sense. If $a$ has components $p+1$ and $1$ with respect to the new system, 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 $p=\frac{-1}{3}$

Here $\overrightarrow{a}=\left(2p\right)\hat{i}+\hat{j}$ when a system is rotated the new component of $\overrightarrow{a}$ are $p+1$ and $1$

$\overrightarrow{b}=(p+1)\hat{i}+\hat{j}$

$\Rightarrow {\left|\overrightarrow{a}\right|}^2={\left|\overrightarrow{b}\right|}^2$

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

$\Rightarrow 4p^2=p^2+2p+1$

$\Rightarrow 3p^2-2p-1=0$

$\Rightarrow (3p+1)(p-1)=0$

$\therefore 3p+1=0,p-1=0$

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