Help With Inverse of 3-d Matrix Rotations Qs b,c,d

[Benny]

I'm having trouble with the following question - parts b, c and d. Can someone please help me out?

Q. 3-d rotations - Consider the matrix:

$$A\left( \theta \right) = \left[ {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } & 0 \\ {\sin \theta } & {\cos \theta } & 0 \\ 0 & 0 & 1 \\ \end{array}} \right]$$

a) Evaluate det(A(theta)).
b) Interpret geometrically the effect of multiplying a vector by A(theta).
c) Show that $$A\left( \theta \right)A\left( \phi \right) = A\left( {\theta + \phi } \right)$$ and interpret this result.
d) Use the previous part to find the inverse of A(theta). How does this compare with the transpose A(theta)^T - the transpose of A(theta).

The answer to part 'a' is 1 which wasn't all that difficult to get. I'm not sure about part b. Ignoring the 3 row and 3rd column I have a 2 by 2 matrix which represents a rotation through an angle of theta in the anti-clockwise direction (in the x-y plane) but I'm not sure how to ineterpret the given 3 by 3 matrix.

I could show the result of part 'c' but again, I don't know how to interpret the result. I think that it might just be the application of A(phi) followed by A(theta).

I have no idea as to how to part 'd.'

Any help would be good thanks.

 

[Galileo]

You already identified it as a rotation if you consider the xy-plane. So you know what it does to the unit vectors x and y. What does the transformation do to the unit z vector?
If you know the interpretation, the rest is easy.

 

[HallsofIvy]

If multiplying a vector by a matrix causes a movement (as rotation), then the inverse matrix causes the opposite movement.