Affine transformations problem

[springo]

Homework Statement

In R³:
T₁ symmetry with respect to x -√3y = 0 & z = 0
T₂ symmetry with respect to the X axis

Find:
The matrices for T₁ and T₂, T₁(T₂) and check that T₁(T₂) is a rotation around a line.

Homework Equations

The Attempt at a Solution

T₂ is:
$$\begin{pmatrix}<br /> {1}&{0}&{0}&{0}\\ <br /> {0}&{1}&{0}&{0}\\ <br /> {0}&{0}&{-1}&{0}\\ <br /> {0}&{0}&{0}&{-1}<br /> \end{pmatrix}$$

The line in T₁ belongs to z = 0 and the image of (0,0,0) is (0,0,0), therefore the image of (0,0,1) is (0,0,-1).

In the plane z = 0, with the basis {O, (√3/2, 1/2), (1/2, -√3/2)} the transformation's matrix is:
$$\begin{pmatrix}<br /> {1}&{0}\\ <br /> {0}&{-1}<br /> \end{pmatrix}$$

So putting this in the canonical basis and all together in one matrix:
$$\begin{pmatrix}<br /> {1}&{0}&{0}&{0}\\ <br /> {0}&{\frac{\sqrt{3}}{2}}&{-\frac{1}{2}}&{0}\\ <br /> {0}&{\frac{1}{2}}&{\frac{\sqrt{3}}{2}}&{0}\\ <br /> {0}&{0}&{0}&{-1}<br /> \end{pmatrix}$$

Then for T₁(T₂) = T₂·T₁ which yields:
$$\begin{pmatrix}<br /> {1}&{0}&{0}&{0}\\ <br /> {0}&{\frac{\sqrt{3}}{2}}&{-\frac{1}{2}}&{0}\\ <br /> {0}&{-\frac{1}{2}}&{-\frac{\sqrt{3}}{2}}&{0}\\ <br /> {0}&{0}&{0}&{1}<br /> \end{pmatrix}$$

I don't know if I'm doing fine so far, and I don't know how to do the end of the problem.

Thanks for your help.

 

[springo]

(Gentle) bump :)

 

[Mark44]

Run a few vectors through your matrix, and see if the output vectors are rotated around a line. The line x = sqrt(3)y seems a likely candidate for the line. The endpoint of each input vector should be the same distance away from the line as the endpoint of the output vector, and the angle of rotation of an input/output vector pair should be the same as that of any other input/output pair.

That's the way I would go. Hope that helps.
Mark