- #1
springo
- 125
- 0
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.
