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 anti-clockwise sense. If a has components p+1 and 1 with respect to the new system, then

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

Answer 1

The correct option is B $p=1$ or $-\frac{1}{3}$

If x,y are the original components; X,Y the new components and $\alpha$ is the angle of rotation, then $x=Xcos\alpha -Ysin\alpha andy=Xsin\alpha +Ycos\alpha$
$\therefore 2p=(p+1)cos\alpha -sin\alpha$ and $1=(p+1)sin\alpha +cos\alpha$
Squaring and adding, we get $4p^2+1=(p+1{)}^2+1$
$\Rightarrow p+1=\pm 2p\Rightarrow p=1$ or $-\frac{1}{3}$.