# If θ is the angle between any two vectors

Question:

If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b}$, then $|\vec{a} \vec{b}|=|\vec{a} \times \vec{b}|$ when $\theta$ isequal to

(A) 0

(B) $\frac{\pi}{4}$

(C) $\frac{\pi}{2}$

(D) $\pi$

Solution:

Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$.

Then, without loss of generality, $\vec{a}$ and $\vec{b}$ are non-zero vectors, so that $|\vec{a}|$ and $|\vec{b}|$ are positive.

$|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$

$\Rightarrow|\vec{a}||\vec{b}| \cos \theta=|\vec{a}||\vec{b}| \sin \theta$                       $[|\vec{a}|$ and $|\vec{b}|$ are positive $]$

$\Rightarrow \tan \theta=1$

$\Rightarrow \theta=\frac{\pi}{4}$

Hence, $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$ when $\theta$ isequal to $\frac{\pi}{4}$.