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Mathematical Expression Editor

We move things around in space.

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

Moving parametric graphs around

We have (at least!) two ways of writing a parametric formula for a curve in
space This second way, using , , and is what we’re going to think about
today.

Compare and contrast the following vector-valued functions in :

Consider vectors:
Verify the following:

(a)

.

(b)

is orthogonal to .

(c)

The lines and both lie on the plane .

Give a vector-valued formula for a circle of radius that lies in the plane
.

Consider the equations:
How many solutions for , , and do you expect to find?

Draw a picture showing the
geometry of this situation.

Consider the equations:
Parameterize a curve giving the solutions to these equations.

Draw a picture showing
the geometry of this situation.

Consider the equations:
Parameterize a curve giving the solutions to these equations.

Draw a picture showing
the geometry of this situation.

Let and be fixed real numbers. Consider the equations:
Give a general solution in terms of and .

Draw a picture showing the geometry of
this situation.

Moving graphs around

We have just worked with vector-valued functions. The intrepid young mathematician
who wishes to further expand their mind, might wish to press-on, and work with
implicit equations as well.

Explain how graphing is related to all vectors such that

Consider , , and . What will the graph of look like? Confirm your answer by using
something like Desmos, GeoGebra, or WolframAlpha.

Consider , , and . What will the graph of look like? Confirm your answer by using
something like Desmos, GeoGebra, or WolframAlpha.

Consider , , and . For any fixed value of , what will the graph of look like? Confirm
your answer for different values of by choosing a reasonable function and using
something like Desmos, GeoGebra, or WolframAlpha.