Find the projection of $\hat{i} + 2 \hat{j} - 2 \hat{k}$ $\;$ on $\;$ $2 \hat{i} - \hat{j} + 5 \hat{k}$
Let $\overrightarrow{a} = \hat{i} + 2 \hat{j} - 2 \hat{k}$ $\;$ and $\;$ $\overrightarrow{b} = 2 \hat{i} - \hat{j} + 5 \hat{k}$
$\begin{aligned}
\text{Projection of } \overrightarrow{a} \text{ on } \overrightarrow{b} & = \dfrac{\overrightarrow{a} \cdot \overrightarrow{b}}{\left|\overrightarrow{b}\right|} \\\\
& = \dfrac{\left(\hat{i} + 2 \hat{j} - 2 \hat{k}\right) \cdot \left(2 \hat{i} - \hat{j} + 5 \hat{k}\right)}{\sqrt{\left(2\right)^2 + \left(-1\right)^2 + \left(5\right)^2}} \\\\
& = \dfrac{2 - 2 - 10}{\sqrt{4 + 1 + 25}} \\\\
& = \dfrac{-10}{\sqrt{30}}
\end{aligned}$