Question

The vectors $a=3i-2j+2k$ and $b=-i-2k$ 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

Answer: (A)

The diagonals of the parallelogram are $\overrightarrow{\alpha }=\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=\overrightarrow{a}-\overrightarrow{b}$ i.e., $\overrightarrow{\alpha }=2i-2j$ and $\overrightarrow{\beta }=\pm (4i-2j+4k)$ 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 $\theta =\frac{3\pi }{4}$