Vtyjkyuk

When does $veca langle vecb, vecc rangle = langle veca, vecb rangle vecc$, where $langle veca, vecb rangle$ is the dot product of $veca$ and $vecb$?

I can see that it holds when $veca$ and $vecc$ are perpendicular to $vecb$, because then $langle veca, vecb rangle = langle vecb, vecc rangle = 0$, but how do I make sure if there are any other possibilities?

If $langleveca,,vecbrangle=0$, we require $veca=0lorlanglevecb,,veccrangle=0$. We can deal with the case $langlevecb,,veccrangle=0$ similarly, so hereafter assume neither is true.

We need $veca$ to be parallel to $vecc$, where I call the zero vector parallel to everything, and indeed if either $veca$ or $vecc$ vanishes the condition holds. Otherwise parallelism can be stated as $veca=kvecc$ for $kne 0$, so the given condition reduces to $kvecclanglevecb,,veccrangle=langle kvecc,,vecbranglevecc$. For a real-valued inner product, this holds for any $vecb,,vecc$. For a complex-valued inner product linear in its first argument, we also need $langlevecb,,veccrangleinmathbbR$. For a complex-valued inner product antilinear in its first argument, we instead have the final requirement $klanglevecb,,veccrangleinmathbbR$.

An interesting identity of vector arithmetic is the "bac cab rule"

$$ veca times (vecb times vecc) =
vecb langle veca, vecc rangle
- vecc langle veca, vecb rangle $$

Thus,

$$veca langle vecb, vecc rangle = langle veca, vecb rangle vecc
qquad text if and only if qquad
vecb times (veca times vecc) = 0
$$
Since, for nonzero $vecv$ and $vecw$, we have that $vecv times vecw = 0$ if and only if they are parallel, and otherwise $vecv times vecw$ is perpendicular to the plane they span, we conclude:

Theorem: We have $veca langle vecb, vecc rangle = langle veca, vecb rangle vecc $ if and only if one (or more) of the following hold:

• one or more of the vectors are zero,


• $vecb$ is perpendicular to the plane spanned by $veca$ and $vecc$,


• $veca$ and $vecc$ are parallel.

We can consider case by case.

1. If $(b,c)neq 0$ and $(a,b)neq 0$, then $a$ and $c$ are parallele.


2. If, say, $(b,c) = 0$, then you have the zero vector on LHS. Thus $(a,b)=0$ or $c=0$.