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 [tex]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 [tex]w=Aw'[/itex]? What is $A^TA$ for an orthogonal matrix?

Ok, so I'll name my vectors differently. However how did you get $v\cdot w=v^Tw$ ?

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)