PARABOLIC MIRRORS

Problem

1. A ray of light parallel to the unit vector hits a plane mirror. If is a unit vector normal to the mirror and points to the front of the mirror, determine the unit vector parallel to the reflected ray.
2. A mirror is in the shape of a piece cutoff from the paraboloid cutoff by a horizontal plane with the inner surface being the reflecting surface. Show that mirror focuses all light rays parallel to the z axis onto a single point on the z-axis i.e a light ray parallel to the z-axis, after reflection by the mirror strikes the z axis at the same point, irrespective of where the initial ray hits the mirror. What is the focus point of this mirror?
3. If the mirror was cutoff by a horizontal plane from the sphere would it still have the above focusing property? Prove your claim.

Suggestions

• The first problem does not require the use of Maple and the final answer is not messy. Try numerical values for and if you are having trouble solving the problem.
• The second problem uses the result from (1) and you may find it convenient to use Maple algebraic and vector manipulation commands.

Instructors : Students need to know that is normal to the surface f(x,y,z)=0. Further, the answer to (1) should be collected first so that those who could not solve (1) may be given the answer to (1)

 
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