Math in Unity : Vectors (Part I)

Before we move on, we need to understand this :
what is a Vector?

If I said that a Vector is a point in a Cartesian plane, would that be right?

For example, if in any 2D project I create a sprite, and put it in this position:

Vector 2, 3, 0 will be the position of the sprite.

BUT there is a difference:
in this case, for example, 2, 3, 0, represents a POINT shifted by 2 on the X axis and 3 on the Y axis, starting from
the ORIGIN POINT, 0, 0, 0.

A Vector, on the other hand, can start from anywhere in space, and its measure will be: 2 on X and 3 on Y.
Furthermore, it will no longer represent a point, but a DIRECTION, and its MAGNITUDE will be given by its length.

It is usually represented like this, with an arrow indicating the direction

How do we calculate the length of a Vector?
With the Pythagorean Theorem.
The formula is this:

|| v ||
is the mathematical notation for the length of a vector.

In this case our equation is
square root of (2 * 2) + (3 * 3)
4 + 9 = 13
square root of 13 is 3.60
and this is the LENGTH of the vector 2, 3, 0.

Having a Vector, and knowing its point of origin, we can calculate its destination simply by making the SUM of the two Vectors.

To give a simple example, if our Vector 2, 3, 0 started from the point
0, 0, 0, its destination will be 2, 3, 0.
If the starting point were 6, 3, 0, its destination will be 6, 3, 0 + 2, 3, 0
the result is 8, 6, 0.

In both cases, the DIRECTION and the LENGTH of the Vector are exactly the same.

I wanted to talk about these little basics regarding the Vectors, because in videogames we will have to deal with them many many times.
It is not necessary to have a complex mathematical background to be able to program video games, but since some formulas or concepts will always be present, I think having a basic idea can be very useful.

In the next articles we will look at the Vectors in more detail, with less theory and more practice in code.