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.