Question

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

• A. $p=0$
• B. $p=0$ or $p=-\frac{1}{3}$
• C. $p=-1$ or $p=\frac{1}{2}$
• D. $p=1$ or $p=-1$

Answer 1

The correct option is D $p=1$ or $p=-1$

The vector $\overrightarrow{a}$ is rotated about the origin $O$.
So even after rotation, the magnitude of the vector will remain the same.
Thus $OA=OB.$
$\therefore$ $\sqrt{4p^2+1}=\sqrt{p^2+2p+1+1}$
$\therefore$ $4p^2=p^2+2p+1$
$\Rightarrow$ $3p^2-2p-1=0.$
$\Rightarrow$ $3p^2-3p+p-1=0.$
$\Rightarrow$ $p=1$ or $p=\frac{-1}{3}$