What is the easiest method to find two vectors that make an orthonormal basis with a given vector?
For example, I have a vector
\begin{pmatrix}
2^{-1/2} \\
-2^{-1/2} \\
0
\end{pmatrix}
I understand that we need to find two unit vectors with cross product 0 with the original vector.
Best Answer
In this case, we can see by inspection that $(0, 0, 1)$ is an acceptable choice. Then, evaluate $(2^{-1/2},-2^{-1/2},0)\times(0,0,1) = (-2^{-1/2},-2^{-1/2},0)$. One can readily check that these are already normalised and so we are done. Perhaps you could have seen what these vectors could have been using geometric considerations as well. But note that our choice of an orthogonal basis is by no means unique. Even if we have two vectors, to complete the orthonormal basis there are still two possible choices.
Alternatively, you could take two linearly-independent vectors and use the Gran-Schmmidt process. There are articles abound on the web mentioning this process.