Question

If $|\vec{a} | = 2 , |\vec{b}| = 7$ and $\vec{a} \times \vec{b} = 3 \hat{i} + 2 \hat{j} + 6\hat{k}$ then the angle between $\vec{a}$ and $\vec{b}$ is

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

Answer 1

Answer: (B)

$|\vec{a} | = 2 , |\vec{b}| = 7$
$\vec{a} \times \vec{b} = 3 \hat{i} + 2 \hat{j} + 6\hat{k}$
we know that $\vec{a} \times \vec{b} = |\vec{a} ||\vec{b}| \sin \theta \hat{n}$
or $| \vec{a} \times \vec{b} | = | |\vec{a}||\vec{b}| \sin \, \theta . \hat{n}|$
$ \Rightarrow \: \sqrt{9 + 4 + 36} = | \vec{a} ||\vec{b}||\sin \, \theta| .1 $
$ \Rightarrow \: \sqrt{49} = 2 \times 7 |\sin \, \theta|$
$\Rightarrow \:\: \frac{1}{2} = \sin \theta \Rightarrow \:\:\: \theta = \frac{\pi}{6}$