Find the unit vector in the direction of the sum of the vectors, $\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$.

$\frac{1}{\sqrt{29}}(4\hat{i} + 3\hat{j} - 2\hat{k})$

The correct answer is Option (2) → $\frac{1}{\sqrt{29}}(4\hat{i} + 3\hat{j} - 2\hat{k})$ ##

The sum of the given vectors is

$\vec{a} + \vec{b} \text{ (= } \vec{c} \text{, say) } = 4\hat{i} + 3\hat{j} - 2\hat{k}$

and $|\vec{c}| = \sqrt{4^2 + 3^2 + (-2)^2} = \sqrt{29}$

Thus, the required unit vector is

$\hat{c} = \frac{1}{|\vec{c}|}\vec{c} = \frac{1}{\sqrt{29}}(4\hat{i} + 3\hat{j} - 2\hat{k}) = \frac{4}{\sqrt{29}}\hat{i} + \frac{3}{\sqrt{29}}\hat{j} - \frac{2}{\sqrt{29}}\hat{k}$