Question

A vector $\overrightarrow{a}$ has components $2$ p 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, $\overrightarrow{a}$ components $p+1$ and $1$, then?

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

Answer 1

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

Note:The magnitude of vector must remain same irrespective of the co-ordinate system.

$⟹\left|\overline{a}\right|=\left|\overline{a^{{}^{\prime }}}\right|$

$|2p\stackrel{\wedge }{i}+\stackrel{\wedge }{j}|=\left|\right(p+1)\stackrel{\wedge }{i^{{}^{\prime }}}+\stackrel{\wedge }{j^{{}^{\prime }}}|$

$⟹\sqrt{{\left(2p\right)}^2+1^2}=\sqrt{{(p+1)}^2+1^2}$

Squaring:

$⟹{4p}^2+1=p^2+1+2p+1$

$⟹{4p}^2+1=p^2+2p+2$

$⟹3p^2-2p-1=0$

$⟹{3p}^2-3p+p-1=0$

$⟹3p(p-1)+1(p-1)=0$

$⟹(p-1)(3p+1)=0$

$⟹p=1,\frac{-1}{3}$