consider a reflection matrix A and a vector x in R2. We define v=x+Ax and w=x-Ax. (v,x,w are vector) a. using the definition of relection, express A(Ax) in terms of X. b.Express Av in terms of v. c.Express Ax in terms of w. d If the vector v and w are both nonzero, what is the angle between v and w. e. If the vector v is nonzero, what is the relationship between v and the line L of relection?

klepkowy7c 2022-07-26 Answered
consider a reflection matrix A and a vector x in R2. We define v=x+Ax and w=x-Ax. (v,x,w are vector)
a. using the definition of relection, express A(Ax) in terms of X.
b.Express Av in terms of v.
c.Express Ax in terms of w.
d If the vector v and w are both nonzero, what is the angle between v and w.
e. If the vector v is nonzero, what is the relationship between v and the line L of relection?
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Answers (1)

Rihanna Robles
Answered 2022-07-27 Author has 18 answers
a. A is a reflection under some line, so A^2 = I, hence A(Ax) = A^2.x = I.x = x, here I is the identity mapping.
b. Av = A(x+Ax) = Ax+A(Ax) = Ax+x (by part a) = x+Ax = v.
c.Aw = Ax-A(Ax) = Ax-x = -(x-Ax) = -w
d.90 degrees
e. parallel

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