$Q$ is an orthogonal matrix, how to prove.
$$\langle Qu,Qv\times Qw\rangle=\langle u,v\times w\rangle$$ for any $u,v,w$ which belong to $\mathbb R^3$
Much obliged if you can help me!
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Does $\times$ mean the cross-product here? – Ben Grossmann Jul 21 '15 at 15:38
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One method is to use the formula proven here – Ben Grossmann Jul 21 '15 at 15:42
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@Omnomnomnom yes ,sorry as I don't know how to use the symbol – cwwwonggg Jul 21 '15 at 15:43
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1$Q$ is just a rotation (or reflection). $\lbrace u, v\times w\rbrace$ is a volume. If you rotate all three vectors, volume doesn't change. Check for reflection. – Michael Galuza Jul 21 '15 at 15:44
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The statement is not true as presented. The correct version is $$ |\langle Qu,Qv\times Qw\rangle| = |\langle u,v\times w\rangle| $$ since the two will have opposite sign if $Q$ is a reflection.
The key here is to note that both quantities give the same volume, or to show that $$ (Qa)\times(Qb) = (\det Q)\; Q (a \times b) $$
Ben Grossmann
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