Reflection Matrices: Verifying Orthogonality and Finding a Unit Vector

In summary, to find a unit vector n such that the line fixed by the reflection is given by the equation n . x = c, you can start by verifying that M(theta) is orthogonal. Then, find the direction of the line of reflection by considering the dot product and the eigenvectors of the matrix. The eigenvalue of the matrix is 1 and as it is a 2x2 matrix, there are expected to be 2 eigenvalues.
  • #1
gomes.
58
0
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.



---------------

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
 
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  • #2
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...
 
  • #3
thanks, how would i find the direction of the line of reflection?

the eigenvalue of the matrix is 1?
 
  • #4
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...
 

FAQ: Reflection Matrices: Verifying Orthogonality and Finding a Unit Vector

What is a reflection matrix?

A reflection matrix is a type of transformation matrix that reflects or flips an object over a line, also known as the line of reflection. It is a 2x2 matrix that can be used to reflect points, lines, or shapes in the plane.

How do you create a reflection matrix?

To create a reflection matrix, you first need to determine the line of reflection. Then, you can use the formula [R] = [I] - 2[u]T[u], where [R] is the reflection matrix, [I] is the identity matrix, and [u] is the unit vector perpendicular to the line of reflection. This formula can be applied to any point, line, or shape in the plane.

What is the effect of a reflection matrix on a shape?

The effect of a reflection matrix on a shape is that it flips or reflects the shape over the line of reflection. This means that the shape will appear as a mirror image of itself on the other side of the line. The distance between the original shape and its reflection will be the same as the distance between the line of reflection and the shape.

Can a reflection matrix be used in 3D space?

Yes, a reflection matrix can be used in 3D space. In this case, the matrix will be a 3x3 matrix and will reflect points, lines, or shapes over a plane instead of a line. The formula for creating a reflection matrix in 3D space is similar to the 2D formula, but it also involves a normal vector for the reflecting plane.

How is a reflection matrix different from a rotation matrix?

A reflection matrix and a rotation matrix are both types of transformation matrices, but they have different effects on shapes. A reflection matrix flips or reflects a shape over a line or plane, while a rotation matrix rotates a shape around a fixed point. In addition, a reflection matrix will change the orientation of the shape, but a rotation matrix will not.

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