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 $-\frac{1}{3}$

$2p$ and $1$ are the components of the original vector whereas $p+1$ and $1$ are the components of the new vector.
Since, they are the same vector rotated about the origin, their magnitude will remain the same.

$\therefore \sqrt{(2p{)}^2+1^2}=\sqrt{(p+1{)}^2+1^2}(2p{)}^2=(p+1{)}^{2}2p=\pm (p+1)\therefore p=1\text{or}p=-\frac{1}{3}$