We study how to bounce vectors off of curves.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Normal vectors

Consider the following line: Find a vector normal to this line. Explain your reasoning.
Now consider the line in . Find a vector normal to this line. Explain your reasoning.
Consider the equation: Explain how this connects to finding normal vectors to lines in of the form: In particular, you should explain what , , and represent.
Quick! Tell me normal vectors for the following lines:
(a)
(b)
(c)
(d)

Reflecting off of lines

Now we will explore how mirrors reflects light.

Let’s see if we can explain why the Law of Reflection is true. We’ll address this in the next several problems.

Consider the following diagram: Explain why the opposite angles and must be equal.
Label more angles in your picture. Some pairs of angles will sum to . Use this to conclude that .

Since a mirror simply reflects light, we see that the initial light beam makes the same angle with the line as the reflected light beam:

Adding a normal vector to the diagram above: Explain why and are equal.
Someone says that if is a normal vector to a mirror, and represents a light beam, then the reflected light beam is given by: Give two explanations verifying the formula above:
  • An explanation where you sketch and show that it is a reasonable answer.
  • An explanation where you show (via a computation) that the angle which light is reflected is the same angle as it arrived, as measured from a line perpendicular to the mirror.
Now imagine the line below as a mirror and imagine vector as a light-ray that strikes the mirror. Find the vector representing the reflection of .

Reflecting off of curves

Reflecting off of mirrored surfaces doesn’t require calculus unless the mirror is curved.

Let We claim that is a normal vector for this curve for any given . Confirm or deny this claim.
Suppose light rays are hitting a mirrored surface described by: Find a formula for the vector reflected off the surface for any given value of . Call this vector . Explain your reasoning.
For any given value of , is a point on our curve. In the previous problem you found a vector describing a reflected light beams . Consider the line:
  • Find so that intersects the -axis.
  • What is the -value of ?
  • Do the questions directly above tell us? What does this have to do with telescopes, space-heaters, vanity mirrors, and eavesdropping devices?