Moving things around

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

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.