Math 1600 Lecture 1, Section 2, 5 Sep 2014

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vector animation

Announcements:

Discuss syllabus. Some key points:

O-week event: "What is Mathematics?" Rescheduled to Friday (today), 3pm-4pm, Middlesex College 107, followed by free pizza and pop!

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 $5-2=3$ and $6-4=2$. 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 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. $[1,2,3,4,5] - [1,0,2,1,1] = [0, 2, 1, 3, 4]$.

\[ \vec{0} := [0, 0, ..., 0] \]