Lecture 1

Chap 1. Vectors, Matrices & Apps. §

$\mathbb{R}$ is the set of all real numbers

Definition of a Vector §

An ordered pair of numbers in ${\mathbb{R}}^2$ e.g. $<2,3>$.

$$<x,y>\text{ where }x\in \mathbb{R},y\in \mathbb{R}$$

An ordered triple of numbers in ${\mathbb{R}}^3$ e.g. $<-1.5,\sqrt{2},\pi >$.

$$<x,y,z>\text{ where }x,y,z\in \mathbb{R}$$

Notation §

in ${\mathbb{R}}^2$:

$$\begin{gathered} \vec{v}=<v_1,v_2>\text{ where }{v}_1,v_2\in \mathbb{R} \\ \vec{v}=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=<v_1,v_2{>}^T\text{ where }{v}_1,v_2\in \mathbb{R} \end{gathered}$$

in ${\mathbb{R}}^3$:

$$\begin{gathered} \vec{v}=<v_1,v_2,v_3>\text{ where }{v}_1,v_2,v_3\in \mathbb{R} \\ =v_1\cdot \vec{i}+v_2\vec{j}+v_2\vec{k}\text{ where }\vec{i}=<1,0,0>,\vec{j}=<0,1,0>,\vec{k}=<0,0,1> \\ i\text{ is the unit vector in the x-direction},j\text{ in the y-direction},k\text{ in the z direction} \end{gathered}$$

Examples §

Let A, B, C, D be 4 points in ${\mathbb{R}}^2$.

$$A=(3,2),B=(-1,4),C=(-4,2),D=(0,0)$$

Point vs Vector Notation

Points are written with Parenthesis e.g $(3,2)$ is a point

Vectors are written with brackets e.g $<3,2>$ is a vector

Consider the vector $\overrightarrow{AB}$ the vector that starts from the point A & ends at the point B.

$$\overrightarrow{AB}=\text{Point }A-\text{Point }B=(-1,4,)-(3,2)=<-4,-2>$$

Length (Magnitude) of a vector §

Def. In 2D §

In ${\mathbb{R}}^2$ the length or magnitude of a vector $\vec{v}=<v_1,v_2>$ is $||\vec{v}||=|\vec{v}|=\sqrt{v_1^2+v_2^2}$

Def. In 3D §

In ${\mathbb{R}}^3$ the length or magnitude of a vector $\vec{w}=<w_1,w_2,w_3>$ is $||\vec{w}||=|\vec{w}|=\sqrt{w_1^2+w_2^2,w_3^2}$

Def. Unit Vector §

We say that a vector $\vec{v}$ in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ is a unit vector of $||\vec{v}||=0$

Direction of a Vector §

In ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ $\vec{v}$ is a vector such that $\vec{v}\ne \vec{0}$.

We say that a vector $\vec{w}$ is in the direction of $\vec{v}$ if there exists :

$$\lambda \in \mathbb{R},\lambda \ge 0\text{ such that }\vec{w}=\lambda \cdot \vec{v}$$

We say that a $\vec{w}$ is in the opposite direction of $\vec{v}$ if there exists:

$$\lambda \in \mathbb{R},\lambda <0\text{ such that }\vec{w}=\lambda \cdot \vec{v}$$

Vector Addition §

$\text{in }{\mathbb{R}}^2:$

$$\vec{v}+\vec{w}=<v_1+w_2,v_2+w_2>$$

$\text{in }{\mathbb{R}}^3:$

$$\vec{v}+\vec{w}=<v_1+w_2,v_2+w_2,v_3+w_3>$$

Vector Multiplication §

Given that $\lambda \in \mathbb{R}$:

$\text{in }{\mathbb{R}}^2:$

$$\lambda \cdot \vec{v}=<\lambda \cdot v_1,\lambda \cdot v_2>$$

$\text{in }{\mathbb{R}}^3:$

$$\lambda \cdot \vec{v}=<\lambda \cdot v_1,\lambda \cdot v_2,\lambda \cdot v_3>$$

Dot Product §

$\text{in }{\mathbb{R}}^2:$

$$\vec{v}\cdot \vec{w}=v_1\cdot w_1+v_2\cdot w_2$$

$\text{in }{\mathbb{R}}^3:$

$$\vec{v}\cdot \vec{w}=v_1\cdot w_1+v_2\cdot w_2+v_3\cdot w_3$$