Question

Column I shows some vector equations. Match Column I with the value of angle between $\overrightarrow{A}and\overrightarrow{B}$ given in Column II.

Column I

Column II

Answer 1

A:
$|\overrightarrow{A}\times \overrightarrow{B}|=|\overrightarrow{A}\cdot \overrightarrow{B}|$
$\Rightarrow AB\sin \theta =AB\cos \theta$
$\Rightarrow \sin \theta =\cos \theta =\sin (\frac{\pi }{2}-\theta ),\sin (\frac{3\pi }{2}-\theta )$
$\Rightarrow \theta =\frac{\pi }{4},\frac{3\pi }{4}$
B:
$\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}=-(\overrightarrow{A}\times \overrightarrow{B})$
$\Rightarrow \overrightarrow{A}\times \overrightarrow{B}=0$
C: $|\overrightarrow{A}+\overrightarrow{B}|=|\overrightarrow{A}-\overrightarrow{B}|$
$\Rightarrow A^2+B^2+2AB\cos \theta =A^2+B^2-2AB\cos \theta$
$\Rightarrow \cos \theta =0$
$\Rightarrow \theta =\frac{\pi }{2}$
D:
$\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}$
$\Rightarrow \sqrt{A^2+B^2+2AB\cos \theta }=C$ .....(1)
$|\overrightarrow{A}|+|\overrightarrow{B}|=|\overrightarrow{C}|$ .....(2)
If $\theta =0$ (1) and (2) will be equal.
A-> 3, A ->4, B ->1, C -> 2, D-> 1