<s>[INST] Products of Orthogonal Vectors Let $a,b,c$ be orthonormal vectors in $\mathbb{R}^n$.  Then is $\sum_{k=1}^n a_{k}b_{k}c_{k} = 0$?  
I tried two examples and it was true, but I don't immediately see why.  
 [/INST] Counterexample:
$$a = \begin{pmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt 3}\\\frac{1}{\sqrt 6}\end{pmatrix},~~b = \begin{pmatrix}\frac{-1}{\sqrt{2}}\\\frac{1}{\sqrt 3}\\\frac{1}{\sqrt 6}\end{pmatrix},~~ c = \begin{pmatrix}0\\\frac{1}{\sqrt 3}\\\frac{-2}{\sqrt 6}\end{pmatrix}.~~$$
Then
$$\sum_k a_kb_kc_k = \frac{2\sqrt 3 - \sqrt 6}{18} \neq 0.$$
Rational counterexample:
$$a = \frac{1}{3} \begin{pmatrix}1\\2\\-2\end{pmatrix},~~b = \frac{1}{3} \begin{pmatrix}2\\1\\2\end{pmatrix},~~c = \frac{1}{3} \begin{pmatrix}-2\\2\\1\end{pmatrix}.$$
In this case, $$\sum_k a_kb_kc_k = \frac{-4}{27} \neq 0.$$
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