Hi, I've got a model solution to this question but am not convinced by it, can anyone help?
Find a matrix for the orthogonal projection such that (1,1,1) and (-1,1,-1) map to the same point.
If these 2 vectors both map to the same point, it must be (0,0,0)
The solution says that (2,0,2) must map to this point, which I figure is because the vector joining (1,1,1) and (-1,1,-1) is a vector that spans the normal to the plane we are projecting onto.
Hence the plane is 2x+2z=0?
Next assuming this is correct I selected 2 vectors that map to themself in the plane and constructed a matrix.
My question is, is there more than one solution depending on the vectors in the plane that are selected?
Also the second part of the question:
Find a matrix for a reflection that maps (1,1,1) to (-1,-1,1)
For this the solution has used the formula 2P-I, where P is the projection matrix from the first part.
This confused me because I thought this formula could only be used if the reflection is around the plane which the matrix P is describing a projection onto.
However if the plane is the same in both parts, then as (1,1,1) is perpendicular to the plane (from the first part), then wouldnt it map to minus itself rather than (-1,-1,1)?