Question

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

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

Answer 1

Answer: (A), (C)

The correct options are
A $\frac{\pi }{4}$
C $\frac{3\pi }{4}$

The two diagonals are given by
$\overrightarrow{AB}-\overrightarrow{BC}=4\hat{i}-2\hat{j}+4\hat{k}$ and $AB+BC=2\hat{i}-2\hat{j}$

These vectors have magnitudes $6$ and $2\sqrt{2}$, respectively and their dot product is $12$.

Therefore the angle between the is
${\cos }^{-1}\frac{12}{\left(6\right)\left(2\sqrt{2}\right)}={\cos }^{-1}\frac{1}{\sqrt{2}}=\frac{\pi }{4}$ or $\pi -\frac{\pi }{4}=\frac{3\pi }{4}$