Question

If $\theta$ is the angle between the vectors $2\overline{i}-2\overline{j}+4\overline{k}$ and $3\overline{i}+\overline{j}+2\overline{k},$ then sin$\theta$ $=$

• A. $\frac{2}{7}$
• B. $\frac{2}{\sqrt{7}}$
• C. $\frac{\sqrt{2}}{7}$
• D. $\sqrt{\frac{2}{7}}$

Answer 1

The correct option is B $\frac{2}{\sqrt{7}}$

$\left(2\hat{i}-2\hat{j}+4\hat{k}\right)\times \left(3\hat{i}+\hat{j}+2\hat{k}\right)$

$=\sqrt{4+4+16}.\sqrt{9+1+4}.\sin \theta$

$=\sqrt{24\times 14}.\sin \theta$

$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -2& 4\\ 3& 1& 2\end{array}\right|$

$=\hat{i}\left(-4-4\right)-\hat{j}\left(4-12\right)+\hat{k}\left(2+6\right)$

$=\left(-8\hat{i}+8\hat{j}+8\hat{k}\right)$

$\therefore \sqrt{64+64+64}=\sqrt{24\times 14}.\sin \theta$

$\therefore \sin \theta =\sqrt{\frac{192}{14\times 24}}$ $=\sqrt{\frac{8}{14}}$

$\sin \theta =\frac{2}{\sqrt{7}}$