Products of Orthogonal Vectors

Question

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.

Answer 1

No, the only conditions you have are $\sum_ka_kb_k=0$ etc. This code finds a random $n=3$ example where $\sum_ka_kb_kc_k\ne0$.

Answer 2

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, $$\sum_k a_kb_kc_k = \frac{-4}{27} \neq 0.$$