$\vec{a} = \lambda\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$, with $\lambda > 0$. For which $\lambda$ is $\vec{a} + \vec{b}$ perpendicular to $\vec{a} - \vec{b}$?

Explanation

1. 01

   $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \lvert \vec{a} \rvert^{2} - \lvert \vec{b} \rvert^{2}$, so the two are perpendicular exactly when $\lvert \vec{a} \rvert = \lvert \vec{b} \rvert$.

2. 02

   $\lambda^{2} + 4 + 1 = 4 + 1 + 9$, so $\lambda^{2} = 9$, and $\lambda > 0$ gives $\lambda = 3$.