Math 1600B Lecture 1, Section 2, 6 Jan 2014
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Announcements:
Discuss syllabus.
Summary of some key points:
 Choice of 1229 vs 1600: 1229 covers less material, is aimed at
social science students, and has fewer prerequisites. But 1600 is
required for many programs. 1229 can be taken before 1600, but
they can't be taken at the same time. See a counsellor if needed.
 This course is cumulative and gets tough. Keep up!
 Before next class, read "To the student", Section 1.0 and Section 1.1
in the text. In general, read the text.
 Do exercises as we cover the
material, and again before quizzes and exams.
 Answers to odd exercises are at the end of the text; solutions are in
the study guide.
 Questions welcome at any time! Are there any now?
New 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 $52=3$ and $64=2$.
We write $\vec{AB} = [3,2] = \coll 3 2$ (order matters).
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 components are equal.
For example, if $C = (3,2)$ and $O = (0,0)$ is the origin, then $\vec{AB} = \vec{OC}$.
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 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 3space, 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. $[1,2,3,4,5] = [1, 2, 3, 4, 5]$.
\[ \vu  \vv := \vu + (\vv) = [u_1v_1, \ldots, u_nv_n] \]
E.g. $[1,2,3,4,5]  [1,0,2,1,1] = [0, 2, 1, 3, 4]$.
\[ \vec{0} := [0, 0, ..., 0] \]
Properties of vector operations:
The picture to the right shows geometrically that vector addition is commutative:
$\vec{u} + \vec{v} = \vec{v} + \vec{u}$.
In this true in $\R^n$? Let's check:
$$
\begin{aligned}
\vec{u} + \vec{v} &= [u_1 + v_1, \ldots, u_n + v_n] \\
&= [v_1 + u_1, \ldots, v_n + u_n] \\
&= \vec{v} + \vec{u}.
\end{aligned}
$$
Many other properties that hold for real numbers also hold for vectors:
Theorem 1.1.
But we'll see differences later.
Example: Simplification of an expression:
$$
\begin{aligned}
&3 \vec{b} + 2 (\vec{a}  4 \vec{b})\\
= &3 \vec{b} + 2 \vec{a}  8 \vec{b}\\
= &2 \vec{a}  5 \vec{b}
\end{aligned}
$$
We'll continue with Section 1.1 in
lecture 2, starting with a video game.