MATH 151 Chapter 1.1

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Introduction to Vectors

Notation

$\mathbf {a} ={\overrightarrow {a}}=\left\langle x,\,y\right\rangle$

...where $x$ and $y$ are components

Definition

2 Properties:

1. Msgnitude
2. Direction

Can be anywhere on 2D plane:

$\left\langle a_{1},\,a_{2}\right\rangle ={\overrightarrow {AB}}\quad A\left(x,\,y\right),\,B\left(x+a_{1},\,y+a_{2}\right)$

If $\left(x,\,y\right)$ is at the origin, then $\left\langle a_{1},\,a_{2}\right\rangle$ is known as position vector.

Vector ${\overrightarrow {AB}}$ between points $A\left(x_{1},\,y_{1}\right)$ and $B\left(x_{2},\,y_{2}\right)$ is given as follows:

$\mathbf {a} =\left\langle x_{2}-x_{1},\,y_{2}-y_{1}\right\rangle$

Magnitude

Also known as length:

$\left|\mathbf {a} \right|={\sqrt {\left(a_{1}\right)^{2}+\left(a_{2}\right)^{2}}}$

Note: $\left|c\mathbf {a} \right|=\left|c\right|\left|\mathbf {a} \right|$

Unit Vectors

Unit vectors are vectors with magnitude of 1:

• $\mathbf {i} =\left\langle 1,\,0\right\rangle$
• $\mathbf {j} =\left\langle 0,\,1\right\rangle$

Regular vectors can be represented as polynomials of unit vectors:

$\left\langle x,\,y\right\rangle =x\mathbf {i} +y\mathbf {j}$

Unit vector in same direction is represented by $\mathbf {u} ={\tfrac {\mathbf {a} }{\left|\mathbf {a} \right|}}$

Operations

Given vectors $\mathbf {a} =\left\langle a_{1},\,a_{2}\right\rangle$ and $\mathbf {b} =\left\langle b_{1},\,b_{2}\right\rangle$ . . .

Addition/Subtraction	${\begin{aligned}\mathbf {a} +\mathbf {b} &=\left\langle a_{1}+b_{1},\,a_{2}+b_{2}\right\rangle \\\mathbf {a} -\mathbf {b} &=\left\langle a_{1}-b_{1},\,a_{2}-b_{2}\right\rangle \end{aligned}}$
Scalar Multiplication	${\begin{aligned}c\mathbf {a} &=\left\langle ca_{1},\,ca_{2}\right\rangle \end{aligned}}$
(more covered later)

Applications

In physics, force can be represented as vectors (magnitude measured in Newtons, and a direction). If several forces are acting on an object, the net resultant force is just the sum of all the smaller forces.