= Solution
Suppose such an $f$ exists. Orthogonality and unit lengths make
$$
H_s(x)=\cos(\pi s)x+\sin(\pi s)f(x),\qquad0\leq s\leq1,
$$
a <continuous map> into the <unit sphere>, since its squared <norm> is one. This is a <homotopy> from the identity to the <antipodal map>. Their degrees would coincide, but for even $n\geq2$ they are respectively $1$ and $(-1)^{n+1}=-1$, a contradiction. Thus \b[no such <continuous map> exists for even $n$]. At $n=0$ there is not even a <unit vector> orthogonal to either point of $S^0\subset\mathbb R$, so nonexistence holds there as well. This is a <sphere> form of the <Hairy ball theorem>.
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