Question

A vector $a$ has components $2p$ and $1$ w.r.t a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If w.r.t the new system, $a$ has components $p+1$ and $1$, then

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

Answer 1

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

Let $i,j$ be unit vectors along the co-ordinate axes

$\therefore \overrightarrow{a}=2pi+1.j$ ....(1)

On rotation, let $b$ be the vector having components $p+1$ and $1$.

$\therefore \overrightarrow{b}=(p+1)i+1.j$ ...(2)

where $i,j$ are units vectors along the new co-ordinate axes.

$\text{But on rotation magnitude of the initial and final vector will remain same}$
$\left|\overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\Rightarrow {\left|b\right|}^2={\left|a\right|}^2$

$\Rightarrow {(p+1)}^2+1={\left(2p\right)}^2+1\Rightarrow 3p^2-2p-1=0$

$\Rightarrow (3p+1)(p-1)=0\Rightarrow p=1$ or $p=-\frac{1}{3}$