Q. A vector $\overrightarrow{\mathbf{a}}$ 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 this respect to new system $\overrightarrow{\mathbf{a}}$ has components $p + 1$ and 1, then
A) $p = 0$
B) $p = 1\ or\ p = \frac{- 1}{2}$
C) $p = - 1$
D) $p = 1\ or\ p = - 1$
Explanation
(b)
Let $,\ \overrightarrow{\mathbf{a}} = 2p\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}}$, $\overrightarrow{\mathbf{b}} = (p + 1)\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}}$
Given, $\left| \ \overrightarrow{\mathbf{a}} \right| = \left| \overrightarrow{\mathbf{b}} \right| \Longrightarrow 4p^{2} + 1 = (p + 1)^{2} + 1$
$\Longrightarrow 3\ p^{2} - 2p - 1 = 0 \Longrightarrow p = 1, - \frac{1}{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.