Question

A vector has component $2p$ and $1$ with respect to a rectangular cartesian system. The axes are rotated through an angle $\alpha$ about the origin in the anticlockwise sense. If the vector has components $p+1$ and $1$ with respect to the new system then

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

Answer 1

The correct option is A $p=1,-\frac{1}{3}$

Even after the vector is rotated through an angle, its magnitude remains the same.
Thus, $(2p{)}^2+1=(p+1{)}^2+1$

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

$\therefore 3p^2-3p+p-1=0$

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

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

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