Cross product proof

Question

Three vectors $a$, $b$, and $c$ are given. Prove that if $a \perp b$, $a \perp c$, $b \perp c$, and $|a|=|b|=|c|=1$, then

$$a=b \times c$$

$$b=c \times a$$

$$c=a \times b$$

Answer 1

Those formulas don't actually always hold. For example if you take the vectors

$$a = \begin{pmatrix}1\\0\\0\end{pmatrix} \quad b=\begin{pmatrix}0\\1\\0\end{pmatrix} \quad c=\begin{pmatrix}0\\0\\-1\end{pmatrix}$$

Then they are orthogonal, but $a \times b = \begin{pmatrix}0\\0\\1\end{pmatrix} =-c$

also $c \times a = -b$ and $b \times c = - a$.