Question:

The angle bisector direction is found by adding the unit vectors of the two given vectors. First, find $|\vec{a}| = |2\hat{i} - 2\hat{j} + \hat{k}| = \sqrt{4+4+1} = 3$ and $|\vec{b}| = |\hat{i} + 2\hat{j} - 2\hat{k}| = \sqrt{1+4+4} = 3$. The bisector direction is $\hat{a} + \hat{b} = \frac{1}{3}(2\hat{i} - 2\hat{j} + \hat{k}) + \frac{1}{3}(\hat{i} + 2\hat{j} - 2\hat{k}) = \frac{1}{3}(3\hat{i} - \hat{k})$. This has magnitude $\frac{1}{3}\sqrt{9+1} = \frac{\sqrt{10}}{3}$. To get magnitude 2, multiply by $\frac{2\sqrt{10}}{10} = \frac{\sqrt{10}}{5}$... Actually, normalize: $\frac{\frac{1}{3}(3\hat{i} - \hat{k})}{\frac{\sqrt{10}}{3}} \times 2 = \frac{2}{\sqrt{10}}(3\hat{i} - \hat{k})$. Option 3 appears to have a typo ($4\hat{i}$ instead of $4\hat{j}$), suggesting both options 1 and 3 may have been intended as the correct answer despite the notation issue.