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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta
name="viewport"
content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"
/>
<title>APMA E2000 - Lines & Planes</title>
<link rel="stylesheet" href="reveal.js/dist/reset.css" />
<link rel="stylesheet" href="reveal.js/dist/reveal.css" />
<link rel="stylesheet" href="reveal.js/dist/theme/dracula.css" />
<link
rel="stylesheet"
href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/all.min.css"
/>
<!-- local styles -->
<link rel="stylesheet" href="css/mvc-slides.css" />
<!-- Theme used for syntax highlighted code -->
<link rel="stylesheet" href="reveal.js/plugin/highlight/monokai.css" />
<link rel="stylesheet" href="plugin/drawer/drawer.css" />
<link
rel="stylesheet"
type="text/css"
href="https://jsxgraph.org/distrib/jsxgraph.css"
/>
<script
type="text/javascript"
src="https://jsxgraph.org/distrib/jsxgraphcore.js"
></script>
</head>
<body>
<div class="reveal">
<div class="slides">
<section>
<section data-auto-animate>
<h3 class="framelabel">
Lecture 03
<a
href="https://excalidraw.com/#room=c29cc686b5dbb8898534,twVMNXhJXLjYdunc8zEvGw"
target="_blank"
rel="noopener noreferrer"
> </a
>
</h3>
<h1>Lines & Planes</h1>
<h2>APMA E2000</h2>
<div class="r-stretch"></div>
<p style="text-align: right">
Drew Youngren <code>[email protected]</code>
</p>
</section>
<section>
<h2>Announcements</h2>
<!-- Set up standard LaTeX macros -->
$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$
$\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$
$\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$
<ul>
<li class="fragment">
Quiz 1 (HW1 topics) this week.
<ul>
<li>HW1 topics</li>
<li>on Gradescope</li>
<li>open for 24 hours (starting Thurs 7pm)</li>
<li>30 minutes window</li>
</ul>
</li>
<li class="fragment">HW2 due next Tues</li>
</ul>
</section>
</section>
<section>
<section>
<h1>1-minute review</h1>
</section>
<section>
<h3>Vector Operations</h3>
<table>
<tr>
<th>Operation</th>
<th>Notation</th>
<th>Formula</th>
</tr>
<tr>
<td>Magnitude</td>
<td>$| \mathbf v |$</td>
<td>$\sqrt{\sum v_i^2}$</td>
</tr>
<tr>
<td>Scalar Multiplication</td>
<td>$c \mathbf v$</td>
<td>$\left\langle c v_1, \ldots, c v_n \right\rangle$</td>
</tr>
<tr>
<td>Vector Addition</td>
<td>$\mathbf v + \mathbf w$</td>
<td>
$\left\langle v_1 + w_1, \ldots, v_n + w_n \right\rangle$
</td>
</tr>
<tr>
<td>Dot Product</td>
<td>$\mathbf v \cdot \mathbf w$</td>
<td>$\sum v_i w_i$</td>
</tr>
<tr class="fragment">
<td>Cross Product</td>
<td>$\mathbf v \times \mathbf w$</td>
<td>
$ \langle v_2 w_3 - v_3 w_2, v_3 w_1 - v_1 w_3, v_1 w_2 - v_2
w_1 \rangle $
</td>
</tr>
</table>
</section>
<section>
<h3>Odds and ends</h3>
<div class="container">
<div class="col">
<div
id="box-lin-combo"
class="jxgbox"
style="width: 400px; height: 400px"
></div>
<script src="js/projection.js"></script>
</div>
<div class="col">
<ul>
<li class="fragment">
$\proj_{\vec w} \vec v = \frac{\vec v \cdot \vec w}{\vec w
\cdot \vec w}\vec w$
</li>
<li class="fragment">
projection breaks vectors into (orthogonal) components
</li>
</ul>
</div>
</div>
</section>
</section>
<section>
<section>
<h1>The Cross Product</h1>
</section>
<section>
<p>
There is a special <strong>vector product</strong>, or
<strong>cross product</strong> in <strong>$\RR^3$ only</strong>.
\[\vec v \times \vec w = \vec u\] "A vector cross a vector is a
vector."
</p>
</section>
<section>
<div class="r-stretch"></div>
<h6 class="framelabel">Defining properties</h6>
<ul class="squarelist">
<li class="fragment">
bilinearity: $ \vec u \times (c\vec v + d\vec w) = c \vec u
\times \vec v + d \vec u \times \vec w$
</li>
<li class="fragment">
skew-symmetry: $\vec v \times \vec w = - \vec w \times \vec v$
</li>
<li class="fragment">
$\vec i \times \vec j = \vec k, \qquad \vec j \times \vec k =
\vec i, \qquad \vec k \times \vec i = \vec j$
</li>
</ul>
<div class="r-stretch"></div>
</section>
<section>
<h3>Formula</h3>
<div class="fragment">
\[\langle a,b,c \rangle\times \langle d,e,f \rangle = (a\vec i +
b\vec j + c \vec k)\times(d\vec i + e\vec j + f \vec k) \]
</div>
<div class="fragment">
\[ = (bf - ce) \vec i + (cd - af) \vec j + (ae - bd) \vec k \]
</div>
<div class="fragment">
$ = \begin{vmatrix} \vec i & \vec j & \vec k \\a &b &c \\ d & e &
f \\ \end{vmatrix} $
<span class="fragment"
>$ \begin{matrix} \vec i & \vec j \\a &b \\ d & e \\
\end{matrix} $</span
>
</div>
<div class="r-stretch"></div>
</section>
<section>
<h3 class="framelabel">Properties of $\vec v \times \vec w$</h3>
<ul>
<li>
$\vec v \times \vec w$ is perpendicular to
<strong>both</strong> $\vec v $ and $\vec w$.
</li>
<li>
$\vec v \times \vec w$ points in the direction according to the
right-hand-rule.
</li>
<li>
$|\vec v \times \vec w| = |\vec v ||\vec w|\sin\theta$ where
theta is the (positive) angle between the vectors.
</li>
</ul>
</section>
<section>
<h3 class="framelabel">Cross Products and Area</h3>
<p>
If $\vec v$ and $\vec w$ define adjacent sides of a parallelogram,
the area is $|\vec v \times \vec w|$.
</p>
<div class="r-stretch">
<iframe
src="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAmc2hvd1BhbmVsPWZhbHNlJm9iajBfa2luZD12ZWN0b3Imb2JqMF9jb2xvcj0lMjMwZDA4ODcmb2JqMF9wYXJhbXNfYT0yJm9iajBfcGFyYW1zX2I9MiZvYmowX3BhcmFtc19jPS0xJm9iajBfcGFyYW1zX3g9MSUyRjImb2JqMF9wYXJhbXNfeT0xJTJGMiZvYmowX3BhcmFtc196PTElMkYyJm9iajBfcGFyYW1zX3QwPTAmb2JqMF9wYXJhbXNfdDE9MSZvYmoxX2tpbmQ9dmVjdG9yJm9iajFfY29sb3I9JTIzZGM1ZTY2Jm9iajFfYW5pbWF0aW9uPXRydWUmb2JqMV9wYXJhbXNfYT0lMjgxKy0rbiUyOSozKyorMSsqK3NpbiUyOHQlMjkrJTJGK3NxcnQlMjg1JTI5KyUyQituKislMjgwLjY2NjY2Nytjb3MlMjh0JTI5Ky0rMC41OTYyODUrc2luJTI4dCUyOSUyOSZvYmoxX3BhcmFtc19iPW4rKislMjgwLjY2NjY2Nytjb3MlMjh0JTI5KyUyQiswLjc0NTM1NitzaW4lMjh0JTI5JTI5Jm9iajFfcGFyYW1zX2M9JTI4MSstK24lMjkqMysqKzIrKitzaW4lMjh0JTI5KyUyRitzcXJ0JTI4NSUyOSsrJTJCK24rKislMjgtMC4zMzMzMzMrY29zJTI4dCUyOSslMkIrMC4yOTgxNDIrc2luJTI4dCUyOSUyOSZvYmoxX3BhcmFtc194PTElMkYyJm9iajFfcGFyYW1zX3k9MSUyRjImb2JqMV9wYXJhbXNfej0xJTJGMiZvYmoxX3BhcmFtc190MD0wJm9iajFfcGFyYW1zX3QxPTIrcGkmb2JqMV9wYXJhbXNfbjA9MCZvYmoxX3BhcmFtc19uMT0xJm9iajJfa2luZD1zdXJmYWNlJm9iajJfY29sb3I9JTIzNGU0YmY3Jm9iajJfYW5pbWF0aW9uPXRydWUmb2JqMl9wYXJhbXNfYT0wJm9iajJfcGFyYW1zX2I9MSZvYmoyX3BhcmFtc19jPTAmb2JqMl9wYXJhbXNfZD0xJm9iajJfcGFyYW1zX3g9MSUyRjIrJTJCKzIrdSslMkIrJTI4MC42NjY2NjcrY29zJTI4dCUyOSstKzAuNTk2Mjg1K3NpbiUyOHQlMjklMjkrKit2Jm9iajJfcGFyYW1zX3k9MSUyRjIrJTJCKzIrdSslMkIrJTI4MC42NjY2NjcrY29zJTI4dCUyOSslMkIrMC43NDUzNTYrc2luJTI4dCUyOSUyOSsqK3Ymb2JqMl9wYXJhbXNfej0xJTJGMistK3UrJTJCKyUyOC0wLjMzMzMzMytjb3MlMjh0JTI5KyUyQiswLjI5ODE0MitzaW4lMjh0JTI5JTI5KnYmb2JqMl9wYXJhbXNfdDA9MCZvYmoyX3BhcmFtc190MT0yK3Bp"
frameborder="0"
width="100%"
height="100%"
></iframe>
</div>
</section>
</section>
<section>
<section>
<h1>Lines</h1>
</section>
<section>
<h3 class="framelabel">Parametric Form</h3>
<p>
A line in $\RR^n$ with position $\vec p$ and direction $\vec v$
has
<strong
><a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
rel="noopener noreferrer"
>parametric form</a
></strong
>
\[ \vec r(t) = \vec p + t\,\vec v \]
</p>
<div class="fragment">
<h6 class="framelabel">Example</h6>
<p>
Find a parametric form for the line through $(-1, 0, -1)$ and
$(1, 1,1)$.
</p>
</div>
</section>
<section>
<h6 class="framelabel">Example</h6>
<p>
Find position and director vectors for the line with parametric
equations: \[ \begin{align*} x &= t - 1 \\ y &= t /2 \\ z &= 3 -
3t \\ \end{align*} \]
</p>
</section>
<section>
<h6 class="framelabel">Example</h6>
<p>
Find the
<a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
rel="noopener noreferrer"
>
intersection
</a>
of the line through $(0,0)$ and $(2,1)$ and the line with position
$\langle 6, -1 \rangle$ and direction $\langle 4, -6 \rangle$.
</p>
<div class="fragment">
<p><strong>Solution.</strong></p>
\[ t \bv{2 \\ 1} = \bv{6 \\ -1} + s \bv{4 \\ -6} \]
<p class="fragment">
$t = 2, s = - \frac{1}{2}$ means intersection point is $(4, 2)$.
</p>
</div>
<p class="fragment">
<strong>Follow up.</strong> At what angle do these lines meet?
</p>
<div class="r-stretch"></div>
</section>
</section>
<section>
<section><h1>Planes</h1></section>
<section>
<h6 class="framelabel">Forms</h6>
<p>
We <em>could</em> define a plane via its
<a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
>parametric form</a
>
with a position vector and 2 direction vectors. \[\vec r(s,t)
=\vec p + s \vec v + t \vec w\]
</p>
<p class="fragment">
But in $\RR^3$, we can define a plane with a 1 position vector
$\vec p$ and one <strong>normal</strong> vector $\vec n$ to define
the
<strong
><a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
rel="noopener noreferrer"
>plane equation</a
></strong
>.
</p>
<div class="fragment">\[\vec n \cdot (\vec x - \vec p) = 0\]</div>
</section>
<section>
<h6 class="framelabel">Exercise</h6>
<p>
What is a normal vector to the
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJm9iajBfa2luZD1sZXZlbCZvYmowX2NvbG9yPSUyMzBkMDg4NyZvYmowX3BhcmFtc19nPXgrJTJCKzIreSslMkIrOCt6Jm9iajBfcGFyYW1zX2s9MTYmb2JqMF9wYXJhbXNfYT0tMiZvYmowX3BhcmFtc19iPTImb2JqMF9wYXJhbXNfYz0tMiZvYmowX3BhcmFtc19kPTImb2JqMF9wYXJhbXNfZT0tMiZvYmowX3BhcmFtc19mPTU="
target="_blank"
rel="noopener noreferrer"
>plane</a
>
given by \[ x+2y = 16-8z \,?\] Find a point on this plane.
</p>
<div class="framelabel"></div>
</section>
<section>
<h6 class="framelabel">Exercises</h6>
<ol>
<li class="fragment">
Find an equation of the
<a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
rel="noopener noreferrer"
>
plane through</a
>
$(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.
<div class="fragment">\[x + y + z = 1\]</div>
</li>
<li class="fragment">
Find a parametric form for the line of intersection of the
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SW50cm8mc2NhbGU9MCZzaG93UGFuZWw9dHJ1ZSZvYmowX2tpbmQ9bGV2ZWwmb2JqMF9jb2xvcj0lMjNmZjI2MDAmb2JqMF9wYXJhbXNfZz14KyUyQit5Ky0reiZvYmowX3BhcmFtc19rPTImb2JqMF9wYXJhbXNfYT0tMyZvYmowX3BhcmFtc19iPTMmb2JqMF9wYXJhbXNfYz0tMyZvYmowX3BhcmFtc19kPTMmb2JqMF9wYXJhbXNfZT0tMyZvYmowX3BhcmFtc19mPTMmb2JqMV9raW5kPWxldmVsJm9iajFfY29sb3I9JTIzMDBmOTAwJm9iajFfcGFyYW1zX2c9Mip4Ky0reSslMkIrMyp6Jm9iajFfcGFyYW1zX2s9MSZvYmoxX3BhcmFtc19hPS0zJm9iajFfcGFyYW1zX2I9MyZvYmoxX3BhcmFtc19jPS0zJm9iajFfcGFyYW1zX2Q9MyZvYmoxX3BhcmFtc19lPS0zJm9iajFfcGFyYW1zX2Y9Mw=="
target="_blank"
>
planes
</a>
given by $x+y-z = 2$ and <br />$2x - y + 3z = 1$.
<div class="fragment">
\[\langle 1 + 2t, 1 - 5t, -3t \rangle\]
</div>
</li>
</ol>
<div class="r-stretch"></div>
</section>
<section>
<ol start="3">
<li>
Find an equation of the plane through the origin and
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJnNjYWxlPTAuNSZzaG93UGFuZWw9ZmFsc2Umb2JqMF9raW5kPWN1cnZlJm9iajBfY29sb3I9JTIzMGQwODg3Jm9iajBfcGFyYW1zX2E9LTImb2JqMF9wYXJhbXNfYj0yJm9iajBfcGFyYW1zX3g9MistK3Qmb2JqMF9wYXJhbXNfeT10Jm9iajBfcGFyYW1zX3o9Mit0KyUyQisxJm9iajBfcGFyYW1zX2EwPTAmb2JqMF9wYXJhbXNfYTE9MQ=="
>the line</a
>
\[\vec r(t) = \bv{2 - t \\ t \\ 2t + 1}.\]
</li>
</ol>
</section>
<section>
<h3 class="framelabel">Distances</h3>
<p>
The distance between sets is defined as the minimum of all
distances between points in the respective sets. \[
\operatorname{dist}(X,Y) = \operatorname{min}\limits_{\vec x \in
X, \vec y \in Y} |\vec x - \vec y| \]
</p>
</section>
<section>
<h6 class="framelabel">Point-to-Plane</h6>
<p>
Find the
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SW50cm8mb2JqMF9raW5kPXBvaW50Jm9iajBfY29sb3I9JTIzZmQyNjI2Jm9iajBfcGFyYW1zX2E9MSZvYmowX3BhcmFtc19iPTEmb2JqMF9wYXJhbXNfYz0yJm9iajBfcGFyYW1zX3QwPTAmb2JqMF9wYXJhbXNfdDE9MSZvYmoxX2tpbmQ9bGV2ZWwmb2JqMV9jb2xvcj0lMjMyNzA1OTImb2JqMV9wYXJhbXNfZz14Ky0reSslMkIrNCt6Jm9iajFfcGFyYW1zX2s9MCZvYmoxX3BhcmFtc19hPS0yJm9iajFfcGFyYW1zX2I9MyZvYmoxX3BhcmFtc19jPS0yJm9iajFfcGFyYW1zX2Q9MyZvYmoxX3BhcmFtc19lPS0yJm9iajFfcGFyYW1zX2Y9MiZvYmoyX2tpbmQ9dmVjdG9yJm9iajJfY29sb3I9JTIzOTkwZDAwJm9iajJfcGFyYW1zX2E9MSZvYmoyX3BhcmFtc19iPS0xJm9iajJfcGFyYW1zX2M9NCZvYmoyX3BhcmFtc194PTAmb2JqMl9wYXJhbXNfeT0wJm9iajJfcGFyYW1zX3o9MCZvYmoyX3BhcmFtc190MD0wJm9iajJfcGFyYW1zX3QxPTEmb2JqM19raW5kPXZlY3RvciZvYmozX2NvbG9yPSUyM2EyMDBmZiZvYmozX3BhcmFtc19hPTEmb2JqM19wYXJhbXNfYj0xJm9iajNfcGFyYW1zX2M9MiZvYmozX3BhcmFtc194PTAmb2JqM19wYXJhbXNfeT0wJm9iajNfcGFyYW1zX3o9MCZvYmozX3BhcmFtc190MD0wJm9iajNfcGFyYW1zX3QxPTEmb2JqNF9raW5kPXZlY3RvciZvYmo0X2NvbG9yPSUyMzAwMDAwMCZvYmo0X3BhcmFtc19hPTErKis0JTJGKzkmb2JqNF9wYXJhbXNfYj0tMSsqKzQlMkYrOSZvYmo0X3BhcmFtc19jPTQrKis0JTJGKzkmb2JqNF9wYXJhbXNfeD0wJm9iajRfcGFyYW1zX3k9MCZvYmo0X3BhcmFtc196PTAmb2JqNF9wYXJhbXNfdDA9MCZvYmo0X3BhcmFtc190MT0xJm9iajVfa2luZD1jdXJ2ZSZvYmo1X2NvbG9yPSUyM2ZmMDBmMiZvYmo1X3BhcmFtc19hPTAmb2JqNV9wYXJhbXNfYj0xJm9iajVfcGFyYW1zX3g9NCUyRjkrJTJCKyUyODErLSs0JTJGOSUyOSp0Jm9iajVfcGFyYW1zX3k9LTQlMkY5KyUyQislMjgxKyUyQis0JTJGOSUyOSp0Jm9iajVfcGFyYW1zX3o9MTYlMkY5KyUyQislMjgyKy0rMTYlMkY5JTI5KnQ="
target="_blank"
rel="noopener noreferrer"
>distance</a
>
from a position $\vec y$ to a plane with normal $\vec n$ and
position $\vec p$.
</p>
<div class="fragment">\[ |\proj_{\vec n} (\vec y - \vec p)| \]</div>
</section>
</section>
<section>
<section><h1>Learning Outcomes</h1></section>
<section id="learning-outcomes">
<h6 class="framelabel">You should be able to...</h6>
<ul>
<li>
Compute and articulate the direction and magnitude of the
cross-product in terms of the inputs.
</li>
<li>
Demonstrate that both lines and planes (in dim 3) can be
determined with 2 vectors.
</li>
<li>
Contrast the (prescriptive) parametric form with the
(descriptive) equation forms of each.
</li>
<li>
Find intersections and generally solve problems of the form
"Find the line plane that..."
</li>
<li>
Select the appropriate role of projection in distance problems
between points, lines, and planes.
</li>
</ul>
</section>
</section>
</div>
</div>
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