# Vectors

A vector is a quantity represented by a direction and a magnitude. Here we will focus mainly on 2d vectors.

now, lets explain what that exactly means, look at this picture with a point P(3,1) on the Cartesian axes.

vector example

you are used to representing point P as P(3,1), but what if you had to refer to it by a direction and magnitude (as in a vector) we can use Pythagoras' theorem to find out the distance to P:

distance = [math]\sqrt{3^2+1^2}[/math] = [math]\sqrt{10}[/math]

The magnitude of P is generally written as |P|, so we have |P| = [math]\sqrt{10}[/math]

we can also use the inverse tan to work out the angle of P from the x axis:

angle: [math]\angle P[/math] = [math]\tan^{-1}({1/3})[/math] = [math]18.43^{\circ}[/math]

The following representation of P is known as the

*of P:*

**Polar**Representation**P**= [math]\sqqrt{10}\angle 18.43^{\circ}[/math]

Sometimes it is more convenient to represent vectors in terms of their x y components and not in terms of an angle and magnitude but in terms of x and y components, you might be asking:

*so then, wasn't P(3, 1) good enough to do this to start with?*

**Unit Vectors**.

A unit vector is simply a vector of magnitude 1, but there are three special unit vectors [math]\hat{i}[/math], [math]\hat{j}[/math] and [math]\hat{k}[/math]. Vectors are sometimes represented with hats or in bold ie

**i**,

**j**and

**k**. Now what are these vectors exactly.

**i**is a vector of magnitude 1 in the direction of the positive x axis,

**j**is a vector of magnitude 1 in the direction of the positive y axis,

**k**is a vector of magnitude 1 in the direction of the positive z axis,

we can now also represent our vector

**P**as: 3

**i**+ 1

**j**+ 0

**k**or:

**P**= 3

**i**+

**j**

*of P. You will get used to representing vectors with both their Polar and Cartesian coordinates.*

**Cartesian**Representationnow you can move on to the next section: Vector Operations