= Vector normalization is not linear
<Vector normalization> is not a <linear map>. With $\mathbf e_1=(1,0)$ and $\mathbf e_2=(0,1)$,
$$
\widehat{\mathbf e_1+\mathbf e_2}
=\frac1{\sqrt2}(1,1)
\ne(1,1)=\widehat{\mathbf e_1}+\widehat{\mathbf e_2}.
$$
Consequently, adding individually normalized directions does not generally give the direction of the original <vector sum>. Compute the intended <vector sum> first, then normalize it if it is nonzero. Cancellation is a further boundary case: $v+(-v)=0$ has no normalized direction even though both nonzero summands do. A <numerical algorithm> must check that case before dividing by the <norm>.
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