If $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$, what is the unit vector in the direction of $\vec{a} + \vec{b}$?

Explanation

1. 01

   $\vec{a} + \vec{b} = 4\hat{i} + \hat{j} + \hat{k}$, with magnitude $\sqrt{16 + 1 + 1} = \sqrt{18}$.

2. 02

   The unit vector is $\frac{1}{\sqrt{18}}(4\hat{i} + \hat{j} + \hat{k})$.