Normalised Vectors

This section explains what a normalised vector is and how it can be used.

Ordinarily, vectors can be used in many situations but sometimes you want to constrain their value (like when dealing with angles) which is why we normalise them. This is essentially a mathematical trick that is used to convert a vector of length n to a vector of length 1, meaning that the vector components get normalised to be between 0 and 1. These vectors are also called unit vectors:

To calculate a normalised vector, we must first have the original vector components, then use them to get the length of the vector. We then divide each of the vector components by this length to get the normalised vector components which form the normalised vector in which the sum of the squares of all coordinates is equal to 1. Here's how:

First we take the coordinates of the vector and get the components:

vx = (x2 - x1) = (7 - 1) = 6
vy = (y2 - y1) = (4 - 1) = 3

We then use these values to calculate the length of the vector:

len = sqr(vx2 + vy2) = sqr(36 + 9) = sqr(45) = 6.708203932499369

Now, that gives us the exact length of the vector a, so let's use that to normalise the two vector components vx and vy:

vx = (vx/len) = (6 / 6.708203932499369) = 0.8944271909999159
vy = (vy/len) = (3 / 6.708203932499369) = 0.4472135954999579
a = 1

Great! We have now normalised the components of the vector! But of what practical use is that in the context of GameMaker: Studio? Okay, let's give you a practical example...

Say you have a game where the player has to shoot at an enemy and you need to know how much the bullet object has to move along the x and y axis each step to hit it:

For this you would use the player and enemy coordinates to get the vector components and the length, then you would normalise them to get a value between 0 and 1 which you would finally multiply by the speed you wish the bullet to travel at each step. These final two values you would then store and add onto the starting x and y coordinates each step. Sound complicated? It's not, look (values have been rounded to one decimal place for simplicity):

px = 100;
py = 425;
ex = 356;
ey = 83;
bullet_speed = 5;

vx = (ex - px) = 256
vy = (ey - py) = -342

len = sqrt(vx2 + vy2) = sqrt(65536 + 116964) = 427.2

vx = vx / len = 0.6
vy = vy / len = 0.8

speed_x = vx * bullet_speed = 3
speed_y = vy * bullet_speed = 4

So, to hit the target we need to add 3 to the bullets x coordinate and 4 to its y coordinate every step.

Back: Vector Functions
Next: What Is A Vector?
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