Linear algebra/matlab: find vector when e2 is reflected over line B with x-axis

[RossH]

Homework Statement

The assignment is to find a matrix, A, that multiplied by a matrix that represents a rectangle will reflect that rectangle about a line through the origin that is at and beta with the x axis. This is all in homogenous coordinates.

Homework Equations

Elementary vectors: e1 is <1,0, ... 0> e2 is <0,1,0 ...> etc.
No others.

The Attempt at a Solution

I thought that the matrix might be: (I hope this formats correctly)
cos(2B) sin(2B) 0
sin(2B) -cos(2B) 0
0 0 1
But when I use this in Matlab, it seems to rotate the rectangle represented by my first matrix in 3 dimensions, rather than reflect it in two dimensions. So right now I am left trying to remake the matrix.

I am trying to figure out the transformation applied to the elementary vectors. When the elementary vector e1 is reflected, that gives me the transformation vector <cos2B,sin2B>
but I am having a lot of trouble figuring out the same vector when e2 is reflected. Any ideas? Thank you.

 

[nate808]

Thank you for your post. It seems like you are on the right track with your initial attempt at a solution. However, I believe the issue may lie in the fact that your matrix is representing a rotation rather than a reflection.

To reflect a point or vector about a line through the origin, we can use the formula:

P' = P - 2(P dot n)n

where P is the original point or vector, P' is the reflected point or vector, and n is the unit normal vector to the reflection line.

In your case, the line is at an angle of beta with the x-axis, so the unit normal vector would be:

n = <cos(beta), sin(beta)>

Using this formula, we can create a reflection matrix A that will reflect any point or vector about this line. It would look like:

A = I - 2nn^T

where I is the identity matrix and T represents the transpose operation.

I hope this helps and please let me know if you have any further questions or need clarification.