Question

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

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

Answer 1

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

$|\overrightarrow{a}\cdot \overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$
$\Rightarrow |\sin \theta |=|\cos \theta |$
$\Rightarrow |\tan \theta |=1$
$\Rightarrow \theta =\frac{\pi }{4},\frac{3\pi }{4}[\because 0\le \theta \le \pi ]$
$\text{But}\overrightarrow{a}\cdot \overrightarrow{b}<0$
$\Rightarrow \frac{\pi }{2}<\theta \le \pi$
Therefore, angle between the vectors is $\theta =\frac{3\pi }{4}$