Consider a circle with radius 1. The vectors OP and OQ shown in the diagram below are the unit vectors i and j rotated by angle t respectively.
OP has coordinates (cos(t), sin(t)) , since the tip of this vector lies on the unit circle, and makes an angle t with the positive x-axis
OQ has coordinates (cos (90 + t), sin (90 + t)) = ( -sin (t), cos (t))
so R [1 0] ^T = [cos(t) sin(t)]^T, ...
A Matrix and Rotating Vectors are investigated.