Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors, such that $\overrightarrow{a}.\overrightarrow{b}<0$ and $|\overrightarrow{a}.\overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$, then angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is

• A. $\pi$
• B. $\frac{7\pi }{4}$
• C. $\frac{\pi }{4}$
• D. $\frac{3\pi }{4}$

Answer 1

The correct option is D $\frac{3\pi }{4}$

We have, $|\overrightarrow{a}.\overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$

$\Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\cos \theta \right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\sin \theta \right|$ $($ where $\theta$ is angle between $\overrightarrow{a}$ and $\overrightarrow{b})$

$\Rightarrow \left|\cos \theta \right|=\left|\sin \theta \right|\Rightarrow \theta =\frac{\pi }{4}$ or $\frac{3\pi }{4}$ $($ as $0\le \theta \le \pi )$

But $\overrightarrow{a}.\overrightarrow{b}<0$ $\therefore \theta =\frac{3\pi }{4}$