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Mathematical Expression Editor

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.

Law of Reflection Light is reflected at the same angle as it arrived, as measured
from a line perpendicular to the mirror. Draw a picture illustrating this
fact.

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?