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Announcements:
These lecture notes
https://jdc.math.uwo.ca/l.
(That's an "ell" at the end.)
Today we start Section 1.3.
Continue reading Section 1.3 and also the Exploration on cross products for
next class.
Work through suggested exercises.
Homework 2 due Friday at 11:55pm.
Math Help Centre:
M-F 12:30-5:30 in PAB48/49 and online 6pm-8pm.
Regular office hours: Mondays, 3:30-4:30, MC130 (Middlesex College)
and Fridays, 2:30-3:30 in the Math Help Centre.
Drop by with any questions!
Partial review of last lecture:
Section 1.2: Length and Angle: The Dot Product
Vectors $\vu$ and $\vv$ are orthogonal if and only if their
dot product $\vu \cdot \vv$ is zero.
Pythagorean theorem in $\R^n$: If $\vu$ and $\vv$ are orthogonal,
then
$$
\| \vu + \vv \|^2 = \| \vu \|^2 + \| \vv \|^2 .
$$
We also saw that $\| \vu + \vv \|^2 = \| \vu \|^2 + 2 \vu \cdot \vv + \| \vv \|^2 $
even if the vectors are not orthogonal.
The projection of $\vv$ onto $\vu$ is given by
$$
\textrm{proj}_{\vu}(\vv) = \left( \frac{\vu \cdot \vv}{\vu \cdot \vu} \right) \vu .
$$
Here $\vu$ must not be $\vec 0$, but $\vv$ can be any vector.
To help remember the formula, note that the denominator ensures that
the answer does not depend on the length of $\vu$.
New material
Section 1.3: Lines and planes in $\R^2$ and $\R^3$
We study lines and planes because they come up directly in applications,
but also because the solutions to many other types of problems can be
expressed using the language of lines and planes.
Lines in $\R^2$ and $\R^3$
Given a line $\ell$, we want to find equations that tell us whether a
point $(x,y)$ or $(x,y,z)$ is on the line.
We'll write $\vx = [x, y]$ or $\vx = [x, y, z]$ for the position vector
of the point, so we can use vector notation.
(Board.) The vector form of the equation for $\ell$ is:
$$
\vx = \vp + t \vd
$$
where $\vp$ is the position vector of a known point on the line,
$\vd$ is a vector parallel to the line, and $t \in \R$.
This is concise and works in $\R^2$ and $\R^3$.
If we expand the vector form into components, we get the
parametric form of the equations for $\ell$:
$$
\begin{aligned}
x\ &= p_1 + t d_1\\
y\ &= p_2 + t d_2\\
( z\ &= p_3 + t d_3 \quad \text{if we are in $\R^3$})
\end{aligned}
$$
Example:
Find the line through the point $P = (2,-1)$ which is
parallel to the vector $\vd = [3,2]$.
Vector form: $\vx = \cyc{ex1-1}{\vp + t \vd} = \cyc{ex1-2}{[2, -1] + t[3,2]}$.
Parametric form: $\cyc{ex1-3}{x = 2 + 3t, \quad y = -1 + 2 t}$.
Example:
Find the line that goes through the points $P = (1,5)$ and $Q = (2,4)$.
First we need to figure out a direction vector $\vd$.
We can take $\vd = \overrightarrow{PQ} = [1,-1].$
Vector form: $\vx = \vp + t \vd = [1, 5] + t[1,-1]$.
Parametric form: $x = 1 + t, \quad y = 5 - t$.
Lines in $\R^2$
There are additional ways to describe a line in $\R^2$.
The normal form of the equation for $\ell$ is:
$$
\vn \cdot (\vx - \vp) = 0 \quad\text{or}\quad \vn \cdot \vx = \vn \cdot \vp ,
$$
where $\vn$ is a vector that is normal = perpendicular to $\ell$.
If we write this out in components, with $\vn = [a, b]$, we get the
general form of the equation for $\ell$:
$$
a x + b y = c,
$$
where $c = \vn \cdot \vp$.
When $b \neq 0$, this can be rewritten as $y = m x + k$, where
$m = -a/b$ and $k=c/b$.
Note: All of these simplify when the line goes through the origin,
as then you can take $\vp = \vec 0$.
Example 5-1: (Board.) Find all four forms of the equations for the line in $\R^2$
going through $A = [1,1]$ and $B = [3,2]$.
Note: None of these equations is unique, as $\vp$, $\vd$ and $\vn$
can all change. The general form is closest to being unique: it is unique
up to an overall scale factor.
(Board: show how equations in Ex 5-1 can change.)
Lines in $\R^3$
Most of the time, one uses the vector and parametric forms above.
But there is also a version of the normal and general forms.
To specify the direction of a line in $\R^3$, it is necessary to
specify two non-parallel normal vectors $\vn_1$ and $\vn_2$.
Then the normal form is
$$
\kern-6ex
\begin{aligned}
\vn_1 \cdot \vx\ &= \vn_1 \cdot \vp
% \qquad{\small\text{typo in book in Table 1.3:}}
\\
\vn_2 \cdot \vx\ &= \vn_2 \cdot \vp
% \qquad{\small\text{there should be no subscripts on $\vp$}}
\end{aligned}
$$
When expanded into components, this gives the general form:
$$
\begin{aligned}
a_1 x + b_1 y + c_1 z\ &= d_1,\\
a_2 x + b_2 y + c_2 z\ &= d_2.
\end{aligned}
$$
Since both equations must be satisfied, this can
be interpreted as the intersection of two planes.
(We'll discuss planes in a second.)
Question: What are the pros and cons of the different ways of
describing a line?
Planes in $\R^3$
Normal form:
$$
\vn \cdot (\vx - \vp) = 0 \quad\text{or}\quad \vn \cdot \vx = \vn \cdot \vp .
$$
This is exactly like the normal form for the equation for a line in $\R^2$.
When expanded into components, it gives the general form:
$$
a x + b y + c z = d,
$$
where $\vn = [a, b, c]$ and $d = \vn \cdot \vp$.
Note: You can read off $\vn$ from the general form.
Two planes are parallel if and only if their normal vectors are parallel.
A plane can also be described in vector form.
You need to specify a point $\vp$ in the plane as well as
two vectors $\vu$ and $\vv$ which are parallel to the plane but not parallel to each other:
$$
\vx = \vp + s \vu + t \vv
$$
When expanded into components, this gives the parametric equations for a plane:
$$
\begin{aligned}
x\ &= p_1 + s u_1 + t v_1\\
y\ &= p_2 + s u_2 + t v_2\\
z\ &= p_3 + s u_3 + t v_3 .
\end{aligned}
$$
Table 1.3 in the text summarizes lines and planes in $\R^3$ nicely:
