Question

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $| \vec{a} . \vec{b}| = | \vec{a} \times \vec{b}|$ , then the angle between $\vec{a}$ and $\vec{b}$ is

• A. $\pi / 4$ only
• B. $\pi / 2$ only
• C. $3\pi / 4$ only
• D. $\pi / 4$ or $3\pi / 4$

Answer 1

Answer: (D)

$\left|\vec{a}.\vec{b}\right|=\left|\vec{a} \times \vec{b}\right| $
$\Rightarrow \left| \left|\vec{a}\right|\left|\vec{b}\right|\cos \theta \right|=\left|\left|\vec{a}\right|\left|\vec{b}\right|\sin \theta \hat{n} \right|$
$\Rightarrow \left|\vec{a}\right|\left|\vec{b}\right|\left|\cos \theta \right| =\left| \left|\vec{a}\right| \left|\vec{b}\right|\sin\theta \hat{n} \right|$
$\Rightarrow \left|\cos \theta\right| = \left|\sin \theta\right|$
$\Rightarrow \theta= \frac{\pi}{4} $ and $\frac{3\pi}{4} $ are possible values