Question

The vectors $\overline{AB}=\overline{3i}-\overline{2j}+\overline{2k}$ and $\overline{BC}=-\overline{i}-\overline{2k}$ are the adjacent sides of a parallelogram. The angle between its diagonals is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$ or $\frac{2\pi }{3}$
• C. $\frac{3\pi }{4}$ or $\frac{\pi }{4}$
• D. $\frac{5\pi }{6}$ or $\frac{\pi }{6}$

Answer 1

Answer: (C)

$\frac{3\pi }{4}$ or $\frac{\pi }{4}$

let diagonals be AC and BD

$\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=2i-2j$

$\overrightarrow{BD}=\overrightarrow{BC}+\overrightarrow{CD}=\overrightarrow{BC}-\overrightarrow{AB}=-4i+2j-4k$

$\therefore \cos \phi =\frac{\overrightarrow{AC}\cdot \overrightarrow{BD}}{\left|\overrightarrow{AC}\right|\left|\overrightarrow{BD}\right|}=\frac{-8-4}{2\sqrt{2}\cdot 6}$

$\therefore \cos \phi =\frac{-1}{\sqrt{2}}$

$\therefore \phi =\frac{\pi }{4}$ or $\frac{3\pi }{4}$