If $\mathbf{a} = \hat{i} + \hat{j} + 2\hat{k}$ and $\mathbf{b} = 2\hat{i} + \hat{j} - 2\hat{k}$, then find the unit vector in the direction of $6\mathbf{b}$.

$\frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k}$

The correct answer is Option (1) → $\frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k}$ ##

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

Since, $6\mathbf{b} = 12\hat{i} + 6\hat{j} - 12\hat{k}$

$∴$ Unit vector in the direction of $6\mathbf{b} = \frac{6\mathbf{b}}{|6\mathbf{b}|}$

$= \frac{12\hat{i} + 6\hat{j} - 12\hat{k}}{\sqrt{12^2 + 6^2 + (-12)^2}} = \frac{6(2\hat{i} + \hat{j} - 2\hat{k})}{\sqrt{324}}$

$= \frac{6(2\hat{i} + \hat{j} - 2\hat{k})}{18} = \frac{2\hat{i} + \hat{j} - 2\hat{k}}{3}$