A laser pointer at point P(0,0,0) emits a beam along the positive x axis. A small, square mirror is located at point M(1.0,0,0) and is free to rotate about both the y (pitch) and z (yaw) axes. The goal is to reflect the beam so that it hits a target on the laboratory ceiling at point T(1.0,2.0,3.0) (coordinates in metres).
(a) Write the condition that must be satisfied by the mirror’s unit normal vector n so that an incident ray from P reflects into a ray toward T. Express your condition symbolically in terms of n and the incident and desired reflected direction vectors.
(b) Solve for a unit normal vector n that meets the requirement in part (a). Show all steps leading to your final numerical components for n.
(c) From the components of n found in part (b), determine the pitch (rotation about the y axis) and yaw (rotation about the z axis) angles the mirror must execute, taking the mirror’s unrotated surface to lie in the yz plane. State each angle to the nearest tenth of a degree.
(d) Briefly explain how the required pitch would change if the target point were moved farther away in the +x direction while keeping its y and z coordinates the same.