In a three dimensional space, consider the point . Rotate this point 45 degrees counter-clockwise about the z-axis (as viewed from above). What are the coordinates of the new point?
Since the rotation is about the z-axis, we are rotating the point (4,3,1) in the plane z=1. Since the rotation occurs in a plane, we can use the two-dimensional counterclockwise rotation matrix
where is the angle of rotation - see Rotation matrix - Wikipedia, the free encyclopedia. If we multiply the vector (T is the transpose) times the rotation matrix (after plugging in a value for ) we will obtain the new x and y coordinates for the rotated point. Since the rotation is in the plane z=1, the z coordinate will still be 1.
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