$\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j}$ and $\vec{c} = \hat{i} - 2\hat{j} + 3\hat{k}$. For which $\lambda$ is $\vec{a} + \lambda\vec{b}$ perpendicular to $\vec{c}$?

Explanation

1. 01

   $(\vec{a} + \lambda\vec{b}) \cdot \vec{c} = \vec{a} \cdot \vec{c} + \lambda\,\vec{b} \cdot \vec{c} = 0$.

2. 02

   $\vec{a} \cdot \vec{c} = 2 - 2 + 6 = 6$ and $\vec{b} \cdot \vec{c} = 1 - 4 + 0 = -3$, so $6 - 3\lambda = 0$ and $\lambda = 2$.