If you’re making your foray into games programming, vectors are an important concept that you’ll have to understand and work with. If you’ve been reading build-your-own-game tutorials and getting confused by all the vector math that’s been going on, this article is probably a good one to read.

### What are vectors?

Remember these diagrams you had to draw in secondary school?

Well, these are vectors. **They represent quantities in space**. This means that **they are like numbers, but they also have a direction**. This makes them effective mathematical concepts for dealing with objects that are moving in 2-dimensional (2D) or 3-dimensional (3D) spaces.

### The Cartesian coordinate system

In a programming context, vectors almost always exist inside a **Cartesian coordinate system**. This is a system of measuring space that uses 2 or 3 number lines to quantify 2D or 3D spaces, respectively.

A number line looks like this:

It is infinite in both directions, and every number is equally spaced from another. **On a number line, you can identify a spot on the line by specifying a number**.

While this may not be very useful, **if you put 2 number lines together, you get a Cartesian coordinate system that can be used to represent 2D space**. In such a system, we call the horizontal line the

**, and the vertical line the**

*x*-axis**.**

*y*-axisAdding a third number line (** z-axis**), as shown below, will give us a

**3D coordinate system**that can quantify 3D space.

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### Vectors in coordinate systems

In a coordinate system, vectors are represented by a set of numbers. **Vectors in a 2D coordinate system contain 2 numbers, while vectors in a 3D coordinate system contain 3 numbers.** No surprises here. Each number in a vector **represents the length of the vector along a given axis**.

When dealing with vectors in Cartesian coordinates, you don’t need vector diagrams to add or subtract vectors. **Simply sum up the numbers on each axis individually:**

**Vectors can also be multiplied or divided;** but unlike addition and subtraction, they can only be multiplied by a single number, or a **scalar**.

Multiplying or dividing a vector **changes its length without changing its direction.** The vector gets longer or shorter, but still points where it used to point.

### Vector lengths

To get the length of a vector, we can apply the **Pythagoras Theorem**. Notice that in a 2D vector, * x* and

*form a right-angled triangle.*

**y**This means that you can get the vector’s length using the following formula:

$\mathit{length}=\sqrt{\mathrm{{x}^{2}+{y}^{2}}}$Pythagoras Theorem works for 3D vectors too:

$\mathit{length}=\sqrt{\mathrm{{x}^{2}+{y}^{2}+{z}^{2}}}$But there are handy shortcuts that you can use in modern game engines to find vector lengths, so you won’t have to apply the formula yourself. **In Unity3D, you can simply use Vector2.magnitude or Vector3.magnitude properties.** These will apply the Pythagoras theorem formula on your vectors to get you the length.

Vector2 v = new Vector(3,5); float dist = v.magnitude;

Vector3 v = new Vector(3,5,2); float dist = v.magnitude;

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