Problem on finding a vector perpendicular to two vectors

Find a unit vector perpendicular to $\langle 1, 1, 1\rangle$ and $\langle 1, 0, 1 \rangle.$
• Solution

Recall that
We identify $\bfx = \langle 1, 1, 1\rangle$ and $\bfy = \langle 1, 0, 1 \rangle$.
Computing the cross product,
Hence, $\bfi - \bfk = \langle 1, 0, -1\rangle$ is a vector that is perpendicular to both $\langle 1, 1, 1\rangle$ and $\langle 1, 0, 0\rangle.$
Let's determine if this vector has unit length.
Recall that
Hence, we observe $\left | \langle 1, 0, -1 \rangle \right | = \sqrt{2}$.
Recall that
Hence, a unit vector that is perpendicular to both $\langle 1, 1, 1\rangle$ and $\langle 1, 0, 0\rangle$ is $$\frac{\langle 1, 0, -1\rangle}{\sqrt{2}} = \left \langle \frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}} \right \rangle.$$