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