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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$2012-11-28

1 Answers 1

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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 $b \times c = - a$.