Question

Find two unit vectors orthogonal to both given vectors. i j k, 4i k

Answer 1

The cross product of two vectors gives a third vector
`\mathbf v` that is orthogonal to the first two.


`\mathbf v=(\vec i+\vec j+\vec k)*(4\,\vec i+\vec k)=\begin{vmatrix}\vec i&\vec j&\vec k\\1&1&1\\4&0&1\end{vmatrix}=\vec i+3\,\vec j-4\,\vec k`

Normalize this vector by dividing it by its norm:


`(\mathbf v)/(\|\mathbf v\|)=(\vec i+3\,\vec j-4\,\vec k)/(√(1^2+3^2+(-4)^2))=\frac1{√(26)}(\vec i+3\,\vec j-4\vec k)`

To get another vector orthogonal to the first two, you can just change the sign and use
`-\mathbf v`.