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- Discuss syllabus and OWL Brightspace page.
- Is 1600 right for you? It's intended for students who require it
for their module. There are many other courses that could fulfill your
first-year math requirements.
For example, 1229, which covers less material.
1229 can be taken before 1600, but they can't be taken at the same time.
See a counsellor if needed.
- 1600 vs 1700: 1700 covers more material and is aimed at math
majors. It is offered in the winter term. New this year.
Can't take both.
New: Time changed so it doesn't conflict with Math 1120B.
- This course is cumulative and gets tough. Keep up!
- Before next class, read "To the student" and Sections 1.0 and 1.1
in the text. In general, read the text as we cover material!
- The OWL Brightspace site has a list of recommended exercises from the text.
Do these exercises as we cover the material, and again before exams.
- Answers to odd exercises are at the end of the text; solutions are in
the study guide, but aren't always clear.
Getting help with course material:
- Help Centre: The Math Help Centre will open in about a week.
We'll announce the schedule when it is available.
- Forum: The Discussions page on OWL Brightspace is a great
place to post questions.
- Office Hours: I will announce my office hours next week.
- Please do not send questions about course content by email, or
come up to the front before or after class.
- Questions are welcome at any time during lecture! Are there any now?
Start of course material
Section 1.1: The Geometry and Algebra of Vectors
| scalar | vector
|
| real valued quantity | magnitude and direction
|
| speed: 10 m/s | velocity: 10 m/s north
$\quad\qquad\begin{CD}{} \\ @AA{10 \text{ m/s}}A \\ {} \end{CD}$
|
| temperature: 10 C | force: 10 Newtons up
$\quad\qquad\begin{CD}{} \\ @AA{10 \text{ N}}A \\ {\smash{\blacksquare}} \end{CD}$
|
| distance: 10 m | displacement: 10 m east
$\quad\lra{\ 10 \text{ m }}\Rule{0pt}{20pt}{0pt}$
|
Vectors in the plane
If $A$ and $B$ are points in the plane, then $\vec{AB}$ denotes
the vector from $A$ to $B$.
The point $A$ is called the initial point
and $B$ is called the terminal point.
(Sketch on board.)
The components of a vector are its horizontal and vertical displacements.
For example, if $A = (2, 4)$ and $B = (5,6)$, then the components of
$\vec{AB}$ are $5-2=3$ and $6-4=2$.
We write $\vec{AB} = [3,2] = \coll 3 2$.
The overall position of a vector does not matter. Two vectors are
considered equal if they have the same length and direction,
or equivalently if their corresponding components are equal.
For example, if $C = (3,2)$ and $O = (0,0)$ is the origin, then $\vec{AB} = \vec{OC}$
since both have components $[3,2]$.
But note that $[2,3]$ is not the same vector, as the order of the components is important.
We write $\R^2$ for the set of all vectors with two real numbers as components.
So $[3,2]$, $[-\pi, 7/2]$ and $\vec 0 = [0,0]$ are all vectors in $\R^2$.
New vectors from old
Vector addition: triangle rule: To add $\vu$ and $\vv$, translate
them so the initial point of $\vv$ equals the terminal point of $\vu$,
and draw an arrow from the initial point of $\vu$ to the terminal point
of $\vv$:
[Drag midpoint to translate vectors, or endpoints to change vectors.
Press "p" to toggle parallelogram rule and "r" to resize canvas.]
Paralleogram rule: Explain with the applet.
Algebraically, to add vectors, you add the corresponding components,
so for $\vu = [u_1, u_2]$ and $\vv = [v_1, v_2]$ we have
\[ \vu + \vv := [u_1+v_1, u_2+v_2] \]
Scalar multiplication: for $c \in \R$ and $\vv = [v_1, v_2]$, we
define
\[ c \vv = c [ v_1, v_2 ] := [c v_1, c v_2 ] . \]
Geometrically, this scales the length by the absolute value $|c|$ of $c$,
and reverses the direction if $c < 0$. (Sketch on board.)
We say that $\vv$ and $\vw$ are parallel if $\vv = c \vw$
or $\vw = c \vv$ for some $c \in \R$.
(Note that $c = 0$ and $c < 0$ are permitted.)
We refer to real numbers as scalars.
Negative: We define $-\vv := (-1)\vv = [-v_1, -v_2]$.
Subtraction: We define $\vu - \vv := \vu + (-\vv) = [u_1 - v_1, u_2 - v_2]$.
Zero vector: We define $\vec{0} = [0, 0]$.
Vectors in $\R^3$
In 3-space, a vector has three components, giving its displacements parallel
to the $x$, $y$ and $z$ axes: $\vv = [v_1, v_2, v_3]$.
The collection of such vectors is denoted $\R^3$.
All of the operations we have discussed extend to $\R^3$.
The text gives some geometrical illustrations.
Vectors in $\R^n$
It is important for applications to be able to deal with vectors
with more than three components.
We write $\R^n$ for the set of ordered $n$-tuples of real numbers.
For example, $[1,0,4,3,2]$ is a vector in $\R^5$.
While we can't visualize such vectors geometrically, the algebraic
definitions extend immediately to this case:
If $\vu = [u_1, u_2, \ldots, u_n]$ and $\vv = [v_1, v_2, \ldots, v_n]$ and
$c \in \R$, then
\[ \vu + \vv := [u_1+v_1,\,\, u_2+v_2,\, \ldots,\,\, u_n+v_n] \]
E.g. $[3, 2, 1, 0] + [1, 0, -1, 4] = [4, 2, 0, 4]$.
\[ c \vu = c [u_1, \ldots, u_n] := [c u_1,\, \ldots,\, c u_n] \]
E.g. $2 [ 1 , -2, 3, -4, 5] = [2, -4, 6, -8, 10]$.
\[ - \vu := (-1) \vu = [-u_1, -u_2, \ldots, -u_n] \]
E.g. $-[1,-2,3,-4,5] = [-1, 2, -3, 4, -5]$.
\[ \vu - \vv := \vu + (-\vv) = [u_1-v_1,\,\, \ldots,\,\, u_n-v_n] \]
E.g. $[1,2,3,4,5] - [1,0,2,1,1] = [0, 2, 1, 3, 4]$.
\[ \vec{0} := [0, 0, ..., 0] \]