Orthogonal transformation problem

[Lunat1c]

Homework Statement

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

\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) for a fixed orthogonal matrix A.

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

I tried to do this by letting

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)

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

 

[gabbagabbahey]

I wouldn't use w=\begin{pmatrix} a_1' \\ a_2' \\ a_3'\end{pmatrix} if I were you, since it is not a general vector, it is related to v=\begin{pmatrix} a_1 \\ a_2 \\ a_3\end{pmatrix} by v=Aw. Instead, you want to use a general vector like w=\begin{pmatrix} b_1 \\ b_2 \\ b_3\end{pmatrix}.

What is v\cdot w=v^Tw if v=Av' and w=Aw'[/itex]? What is A^TA for an orthogonal matrix?

 

[Lunat1c]

I wouldn't use w=\begin{pmatrix} a_1' \\ a_2' \\ a_3'\end{pmatrix} if I were you, since it is not a general vector, it is related to v=\begin{pmatrix} a_1 \\ a_2 \\ a_3\end{pmatrix} by v=Aw. Instead, you want to use a general vector like w=\begin{pmatrix} b_1 \\ b_2 \\ b_3\end{pmatrix}.

What is v\cdot w=v^Tw if v=Av' and w=Aw'[/itex]? What is A^TA for an orthogonal matrix?

<br /> <br /> Ok, so I&#039;ll name my vectors differently. However how did you get v\cdot w=v^Tw ?<br /> <br /> Also, in reply to your question A^TA=I

 

[lanedance]

v\cdot w=v^Tw
this is just a dot product written in matrix notation, the transpose comes about as to satisy matrix multiplication, you must multiply a 1xn matrix (row vector) with a nx1 matrix (column vetcor)