A vector a has components 2p and 1 with respect to a rectangular … - Vidyadip

MCQ

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
$p = 0$
✓
$p = 1$ or $ - \frac{1}{3}$
C
$p = - 1$ or $\frac{1}{3}$
D
$p = 1or - 1$

✓

Answer

Correct option: B.

$p = 1$ or $ - \frac{1}{3}$

b
(b) If $x,\,\,y$ are the original components; $X,\,\,Y$the new components and $\alpha $ is the angle of rotation, then $x = X\cos \alpha - Y\sin \alpha $ and $y = X\sin \alpha + Y\cos \alpha $

$\therefore \,2p = (p + 1)\cos \alpha - \sin \alpha $ and $1 = (p + 1)\sin \alpha + \cos \alpha $

Squaring and adding, we get $4{p^2} + 1 = {(p + 1)^2} + 1$

$ \Rightarrow p + 1 = \pm {\rm{ }}2p \Rightarrow p = 1$ or $ - \frac{1}{3}.$

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