Question

If $|\overrightarrow{a}\cdot \overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|,$ then the acute angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is
$\left(a\right){30}^{\circ }\left(b\right){45}^{\circ }$
$\left(c\right){60}^{\circ }\left(d\right){90}^{\circ }$

Answer 1

We know that,
$\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \theta$
$\overrightarrow{a}\times \overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\sin \theta \hat{n}$

Now,
$|\overrightarrow{a}.\overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$
$\Rightarrow |\overrightarrow{a}||\overrightarrow{b}|\cos \theta =|\overrightarrow{a}||\overrightarrow{b}|\sin \theta$
$\Rightarrow \sin \theta =\cos \theta$
$\Rightarrow \tan \theta =1$
$\Rightarrow \theta ={45}^{\circ }$
Ans : $\left(b\right)$