If (\vec{a} = 4\hat{i} - \hat{j} + \hat{k}) and (\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k}), then find a unit vector parallel to the vector (\vec{a} + \vec{b})

(\vec{a} = 4\hat{i} - \hat{j} + \hat{k}) and (\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k})
(\vec{a} + \vec{b} = (4+2)\hat{i} + (-1-2)\hat{j} + (1+1)\hat{k} = 6\hat{i} - 3\hat{j} + 2\hat{k})
Unit vector parallel to (\vec{a} + \vec{b}) is given by:
(\frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|} = \frac{6\hat{i} - 3\hat{j} + 2\hat{k}}{\sqrt{6^2 + (-3)^2 + 2^2}} = \frac{6\hat{i} - 3\hat{j} + 2\hat{k}}{\sqrt{36 + 9 + 4}} = \frac{6\hat{i} - 3\hat{j} + 2\hat{k}}{\sqrt{49}} = \frac{6\hat{i} - 3\hat{j} + 2\hat{k}}{7})
Therefore, the unit vector parallel to (\vec{a} + \vec{b}) is (\frac{6}{7}\hat{i} - \frac{3}{7}\hat{j} + \frac{2}{7}\hat{k}).