Find the sine of the angle between the vectors $\mathbf{a} = 3\hat{\mathbf{i}} + \hat{\mathbf{j}} + 2\hat{\mathbf{k}}$ and $\mathbf{b} = 2\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 4\hat{\mathbf{k}}$.

$\frac{2}{\sqrt{7}}$

The correct answer is Option (3) → $\frac{2}{\sqrt{7}}$ ##

Given vectors are

$\mathbf{a} = 3\hat{\mathbf{i}} + \hat{\mathbf{j}} + 2\hat{\mathbf{k}}$ and $\mathbf{b} = 2\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 4\hat{\mathbf{k}}$

Let $\theta$ be the angle between the vector $\mathbf{a}$ and $\mathbf{b}$ then $\sin \theta = \frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}| |\mathbf{b}|}$

$∴\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 3 & 1 & 2 \\ 2 & -2 & 4 \end{vmatrix} = \hat{\mathbf{i}}(4 + 4) - \hat{\mathbf{j}}(12 - 4) + \hat{\mathbf{k}}(-6 - 2)$

$= 8\hat{\mathbf{i}} - 8\hat{\mathbf{j}} - 8\hat{\mathbf{k}}$

$|\mathbf{a} \times \mathbf{b}| = \sqrt{(8)^2 + (8)^2 + (-8)^2} = \sqrt{64 + 64 + 64} = \sqrt{192}$

$∴\sin \theta = \frac{\sqrt{192}}{\sqrt{14} \times \sqrt{24}} = \frac{8\sqrt{3}}{\sqrt{8} \times \sqrt{3} \times \sqrt{2} \times \sqrt{7}} = \frac{\sqrt{8}}{\sqrt{2} \times \sqrt{7}} = \frac{\sqrt{4}}{\sqrt{7}} = \frac{2}{\sqrt{7}}$

$[∵|\mathbf{a}| = \sqrt{3^2 + 1^2 + 2^2} = \sqrt{14}, |\mathbf{b}| = \sqrt{2^2 + (-2)^2 + (4)^2} = \sqrt{24}]$