a) Prove that ||B(x, y)|| = ||(x, y)|| for all (x, y) R2. b) Let (x, y) R2 be a nonzero vector and

a) Prove that ||B(x, y)|| = ||(x, y)|| for all (x, y) ˆˆ R2.
b) Let (x, y) ˆˆ R2 be a nonzero vector and φ represent the angle between B(x, y) and (x, y). Prove that cos φ = cos θ. Thus, show that B rotates R2 through an angle θ. (When θ > 0, we shall call B counterclockwise rotation about the origin through the angle 0.)
Let

for some 0 ˆˆ R.

sin cos