How can you find a reflection matrix about a given

Celia Horne 2022-01-31 Answered
How can you find a reflection matrix about a given line, using matrix multiplication and the idea of composition of transformations?
The line of: y=2x3, all in R2
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dikgetse3u
Answered 2022-02-01 Author has 10 answers
Step 1
So you have a vector (x1,y1)T, and you want to find a matrix M such that (x2,y2)T=M(x1,y1)T is the reflection across y=2x3.
So what does the reflection means? It means that the middle of the two points is on the line of reflection, and the line between those points is perpendicular to the reflection line. The first condition can be written as
y1+y22=23x1+x22
The second condition means that the slope of the line between the two points is 1m where m is the slope of the reflection line:
y2y1x2x1=123=32
You now write x2 and y2 in terms of x1 and y1:
x2=m11x1+m12y1
y2=m21x1+m22y1
Then the matrix you are looking for has these coefficients.
M=(m11m12m21m22)
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Karly Logan
Answered 2022-02-02 Author has 11 answers
Step 1
One strategy is to rotate the plane so that the line becomes y=0, apply the reflection in the x-axis, and then rotate back.
Let tanθ=23. Then the required matrix is
(cosθsinθsinθcosθ)(1001)(cosθsinθsinθcosθ)
=(cos2θsin2θsin2θcos2θ)
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