Given a vector $\mathbf{a}$ I want to transform the standard basis $\{\mathbf{e}_1, \dots, \mathbf{e}_n\}$ so that $$ \mathbf{e}_1 \mapsto \mathbf{a} $$ while the remaining vectors are rotated so that they again form an orthogonal basis. Another way to state this is that I want to rotate the hyperplane orthogonal to $\mathbf{e}_1$ so that it is now orthogonal to $\mathbf{a}$. Now there are many linear transformations that accomplish this. But intuition suggests that there ought to be a simplest such transformation which doesn't also mirror the plane or any other funky business. Does somebody have a suggestion?
2026-05-05 13:35:53.1777988153
How do you rotate the standard basis to line up with a given vector?
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