Question

The vectors $\overrightarrow{a}=3\hat{i}-2\hat{j}+2\hat{k}$ and $\overrightarrow{b}=-\hat{i}-2\hat{k}$ are the adjacent sides of a parallelogram.Then,angle between its diagonals is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is A $\frac{\pi }{4}$

The diagonals of the parallelogram are given by
$\overrightarrow{\alpha }=\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=\pm (\overrightarrow{a}-\overrightarrow{b})$
$\Rightarrow \overrightarrow{\alpha }=2\hat{i}-2\hat{j}$ and $\overrightarrow{\beta }=\pm (4\hat{i}-2\hat{j}+4\hat{k})$
Let $\theta$ be the angle between the diagonals. Then,
$Cos\theta =\frac{\overrightarrow{\alpha }.\overrightarrow{\beta }}{|\overrightarrow{\alpha }||\overrightarrow{\beta }|}$
$\Rightarrow cos\theta =\frac{1}{\sqrt{2}}$ or $cos\theta =-\frac{1}{\sqrt{2}}$
$\Rightarrow \theta =\frac{\pi }{4}$ or $\theta =\frac{3\pi }{4}$