Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors, such that $\overrightarrow{a}\cdot \overrightarrow{b}<0$ and $|\overrightarrow{a}\cdot \overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$, then the 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 $|\overrightarrow{a}\cdot \overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$
$\Rightarrow |\overrightarrow{a}||\overrightarrow{b}||\cos \theta |=|\overrightarrow{a}||\overrightarrow{b}||\sin \theta |$
(where $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$)

$\Rightarrow |\cos \theta |=|\sin \theta |$
$\Rightarrow \theta =\frac{\pi }{4}$ or $\frac{3\pi }{4}(as0\le \theta \le \pi )$

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