We move things around in space.

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

1 Moving parametric graphs around

We have (at least!) two ways of writing a parametric formula for a curve in space

\[ \vector {x(t),y(t),z(t)}\quad \text {and}\quad x(t) \veci + y(t) \vecj + z(t) \veck \]

This second way, using \(\veci \), \(\vecj \), and \(\veck \) is what we’re going to think about today.

Compare and contrast the following vector-valued functions in \(\R ^3\):
\begin{align*} \vec {a}(t) &= \veci \cos (t) + \vecj \sin (t)\\ \vec {b}(t) &= \veci \cos (t) + \veck \sin (t)\\ \vec {c}(t) &= \vecj \cos (t) + \veck \sin (t) \end{align*}
Consider vectors:
\begin{align*} \uvec {v} &= \vector {\frac {1}{\sqrt {6}},\frac {-2}{\sqrt {6}},\frac {1}{\sqrt {6}}}\\ \uvec {w} &= \vector {\frac {1}{\sqrt {2}},0,\frac {-1}{\sqrt {2}}} \end{align*}

Verify the following:

(a)
\(|\uvec {v}| = |\uvec {w}| = 1\).
(b)
\(\uvec {v}\) is orthogonal to \(\uvec {w}\).
(c)
The lines \(\vecl (t) = t \uvec {v}\) and \(\vec {m}(t) = t \uvec {w}\) both lie on the plane \(x+y+z = 0\).
Give a vector-valued formula for a circle of radius \(1\) that lies in the plane \(x+y+z=0\).
Consider the equations:
\begin{align*} x+y &= 6\\ x^2+y^2+z^2 &= 18 \end{align*}

How many solutions for \(x\), \(y\), and \(z\) do you expect to find?

Draw a picture showing the geometry of this situation.
Consider the equations:
\begin{align*} x+y &= 0\\ x^2+y^2+z^2 &= 18 \end{align*}

Parameterize a curve giving the solutions to these equations.

Draw a picture showing the geometry of this situation.
Consider the equations:
\begin{align*} x+y &= 2\\ x^2+y^2+z^2 &= 18 \end{align*}

Parameterize a curve giving the solutions to these equations.

Draw a picture showing the geometry of this situation.
Let \(a\) and \(b\) be fixed real numbers. Consider the equations:
\begin{align*} x+y &= a\\ x^2+y^2+z^2 &= b \end{align*}

Give a general solution in terms of \(a\) and \(b\).

Draw a picture showing the geometry of this situation.

2 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
\[ y= x^2 \]
is related to all vectors \(\vec {x}= \vector {x,y}\) such that
\[ \scal _\vecj (\vec {x}) = \scal _\veci (\vec {x})^2. \]
Consider \(\vec {v} = \vector {1,1}\), \(\vec {w} = \vector {-1,1}\), and \(\vec {x} = \vector {x,y}\). What will the graph of
\[ \scal _\vec {w}(\vec {x}) = \scal _\vec {v}(\vec {x})^2 \]
look like? Confirm your answer by using something like Desmos, GeoGebra, or WolframAlpha.
Consider \(\vec {v} = \vector {1,-2}\), \(\vec {w} = \vector {2,1}\), and \(\vec {x} = \vector {x-3,y+4}\). What will the graph of
\[ \scal _\vec {w}(\vec {x}) = \scal _\vec {v}(\vec {x})^2 \]
look like? Confirm your answer by using something like Desmos, GeoGebra, or WolframAlpha.
Consider \(\uvec {v} = \vector {\cos (\theta ),\sin (\theta )}\), \(\uvec {w} = \vector {-\sin (\theta ),\cos (\theta )}\), and \(\vec {x} = \vector {x,y}\). For any fixed value of \(\theta \), what will the graph of
\[ \scal _\uvec {w}(\vec {x}) = f(\scal _\uvec {v}(\vec {x})) \]
look like? Confirm your answer for different values of \(\theta \) by choosing a reasonable function \(f\) and using something like Desmos, GeoGebra, or WolframAlpha.