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

Here, $\overrightarrow{a}=\left(2p\right)\hat{i}+\hat{j},$
When the system is rotated, the new components of $\overrightarrow{a}$ are (p + 1) and 1.
i.e. $\overrightarrow{b}=(p+1)\hat{i}+\hat{j}$
But $|\overrightarrow{a}{|}^2=|\overrightarrow{b}{|}^2$
$\Rightarrow 4p^2+1=(p+1{)}^2+1\Rightarrow 4p^2=p^2+2p+1\Rightarrow 3p^2-2p-1=0\Rightarrow (3p+1\left)\right(p-1)=0\Rightarrow p=1,-1/3$