Reflection Matrices: Verifying Orthogonality and Finding a Unit Vector

[gomes.]

Verify that M(theta) is orthogonal, and find a unit vector n such that the line fixed by the reflection is given by the equation

n . x = c,

for a suitable constant c, which should also be determined.



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I did the verficiation part, by multiplying m(theta) by its transpose. But how do I do the 2nd part? (regarding the find a unit vector).

[PLAIN]http://img268.imageshack.us/img268/4686/123wrm.jpg

 

[lanedance]

how about starting by finding the direction of the line of reflection...

then using the info you find, think about the dot product

you could also consider the eigenvectors of the matrix as well...

 

[gomes.]

thanks, how would i find the direction of the line of reflection?

the eigenvalue of the matrix is 1?

 

[lanedance]

do you have any ideas how to do it, or have you tried anything ?

as its a 2x2 matrix I would expect it to have 2 eigenvalues...