Question

If $a$ and $b$ unit vectors such that $[abc]=\frac{1}{4}$, then angle between $a$ and $b$ is?

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{6}$
• D. $\frac{\mathrm{\pi }}{2}$
• E. $\frac{\mathrm{\pi }}{12}$

Answer 1

The correct option is C

$\frac{\mathrm{\pi }}{6}$

Explanation for the correct option:

Find the angle between $a$ and $b$:

Given, $[abc]=\frac{1}{4}$

$\Rightarrow$ $[a\times b,a\times b]=\frac{1}{4}$

$\Rightarrow$$(a\times b).(a\times b)=\frac{1}{4}$

$\Rightarrow$ $|axb{|}^2=\frac{1}{4}$

$\Rightarrow$ $|axb|=\frac{1}{2}$

$\Rightarrow$ $a.b\sin \theta =\frac{1}{2}$ $\left[\mathbf{\because }\left|a\times b\right|\mathbf{}\mathbf{=}\mathbf{}\mathit{a}\mathbf{.}\mathit{b}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }\right]$

$\Rightarrow$ $sin\theta =\frac{1}{2}$ [ $\because a$and $b$ are unit vectors]

$\therefore \theta =\frac{\mathrm{\pi }}{6}$

Hence, Option ‘C’ is Correct.