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Lets say I fix 3 mutually orthogonal unit vectors i, j and k. Consider an orthogonal transformation F of vectors defined by [tex] F(a_1i+ a_2j + a_3k)=a_1'i+a_2'k+a_3'k [/tex] where

[tex] \left( \begin{array}{ccc} a_1 \\ a_2 \\ a_3\end{array}\right) = A\left( \begin{array}{ccc} a_1' \\ a_2' \\ a_3'\end{array}\right) [/tex] for a fixed orthogonal matrix A.

How can I show that F(v).F(w)=v.w?

I tried to do this by letting

[tex] v = \left( \begin{array}{ccc} a_1 \\ a_2 \\ a_3\end{array}\right), w= \left( \begin{array}{ccc} a_1' \\ a_2' \\ a_3'\end{array}\right) [/tex]

but the fact that I don't know A is holding me back from doing this, so I think there must be some other approach.

Any help would be much appreciated

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# Homework Help: Orthogonal transformation problem

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