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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
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 Polar Representation of P:
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?
Well, it does represent the point in terms of x and y components however having P(3,1) inside an equation makes it much more confusing to read. so I will now introduce some 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: 3i + 1j + 0k or:
P = 3i + j
This is known as the Cartesian Representation of P. You will get used to representing vectors with both their Polar and Cartesian coordinates.

now you can move on to the next section: Vector Operations