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