I need to find the matrix of reflection through line y=-(2)/(3)x

Lilliana Livingston 2022-07-16 Answered
I need to find the matrix of reflection through line y = 2 3 x
I'm trying to visualise a vector satisfying this. The standard algorithm states that we need to find the angle this line makes with x axis and the transformation matrix can be seen as R α T 0 R α .
I'm not sure how to proceed. I can't visualise the angle it makes with x axis. Is there a procedure to think about such reflections?
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Answers (2)

eri1ti0m
Answered 2022-07-17 Author has 11 answers
Alternatively to the comment above (which requires a bit of trig), where does your matrix T send ( 3 2 ) and ( 2 3 ) which are on and perpendicular to your line respectively. Now can you find a and b in terms of x and y so that
( x y ) = a ( 3 2 ) + b ( 2 3 )
Finally, apply T to find T ( x y )
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Avery Stewart
Answered 2022-07-18 Author has 2 answers
Notice that vectors on this line have the form ( 1 , 2 3 ), and an orthogonal vector would be ( 2 3 , 1 ). A very straightforward procedure could be to reflect the vectors ( 0 , 1 ) and ( 1 , 0 ) orthogonally in this line. Once you have determined the images of the basis vectors, you can figure out what the matrix columns should look like.
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