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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

$$a=b \times c$$

$$b=c \times a$$

$$c=a \times b$$

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This does not seem to be true, e.g. taking $e_1,e_2,e_3 \in \mathbb{R}^3$, then $e_2 \times e_3 = -e_1$ instead of $e_1$ – anegligibleperson Nov 28 '12 at 16:51
up vote 3 down vote accepted

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 $b \times c = - a$.

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