Why does
multiplying points by R rotate them?
• Think of the
rows of R as a new coordinate system. Taking inner products of each points with these
expresses that point in that coordinate
system.
• This means rows
of R must be orthonormal vectors (orthogonal
unit vectors).
• Think of what
happens to the points (1,0) and (0,1).
They go to (cos theta, -sin theta), and
(sin theta, cos theta). They remain orthonormal, and rotate clockwise by theta.
• Any other
point, (a,b) can be thought of as a(1,0) + b(0,1).
R(a(1,0)+b(0,1) = Ra(1,0) + Ra(0,1) = aR(1,0) + bR(0,1). So
it’s in the same position relative to the rotated coordinates that it was in before rotation
relative to the x, y coordinates. That is, it’s rotated.