What is the projection vector of $\vec{a} = 3\hat{i} + \hat{j} - \hat{k}$ on $\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}$?

Explanation

1. 01

   The projection vector of $\vec{a}$ on $\vec{b}$ is $\left(\frac{\vec{a} \cdot \vec{b}}{\lvert \vec{b} \rvert}\right)\hat{b} = \frac{\vec{a} \cdot \vec{b}}{\lvert \vec{b} \rvert^{2}}\vec{b}$.

2. 02

   $\vec{a} \cdot \vec{b} = 3 + 2 - 2 = 3$ and $\lvert \vec{b} \rvert^{2} = 9$, so it is $\frac{3}{9}\vec{b} = \frac{1}{3}(\hat{i} + 2\hat{j} + 2\hat{k})$.