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.