Answer: The transformation matrix R for a rotation of 120° about an axis from the origin through the point (1,1,1) can be calculated using the formula:
R = I + (sin(θ))U + (1-cos(θ))UU'
where θ is the angle of rotation (120° in this case), I is the identity matrix, U is a skew-symmetric matrix, and U' is its transpose. The skew-symmetric matrix can be calculated as follows:
U = [0, -z, y; z, 0, -x; -y, x, 0]
where x, y, and z are the components of the vector (1,1,1) that defines the axis of rotation.
Plugging in the values and simplifying, the transformation matrix R is:
R = [ cos(120°), -sin(120°), 0; sin(120°), cos(120°), 0; 0, 0, 1 ]
Note: cos(120°) = -1/2 and sin(120°) = √(3)/2.
Explanation: