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

$\frac{1}{\sqrt{37}}(\hat{j} + 6\hat{k})$

The correct answer is Option (2) → $\frac{1}{\sqrt{37}}(\hat{j} + 6\hat{k})$ ##

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

Since, $2{a} - {b} = 2(\hat{i} + \hat{j} + 2\hat{k}) - (2\hat{i} + \hat{j} - 2\hat{k})$

$= 2\hat{i} + 2\hat{j} + 4\hat{k} - 2\hat{i} - \hat{j} + 2\hat{k} = \hat{j} + 6\hat{k}$

$∴$ Unit vector in the direction of $2{a} - {b} = \frac{2{a} - {b}}{|2{a} - {b}|} = \frac{\hat{j} + 6\hat{k}}{\sqrt{1 + 36}} = \frac{1}{\sqrt{37}}(\hat{j} + 6\hat{k})$