#### Overview of this book

Physics is really important for game programmers who want to add realism and functionality to their games. Collision detection in particular is a problem that affects all game developers, regardless of the platform, engine, or toolkit they use. This book will teach you the concepts and formulas behind collision detection. You will also be taught how to build a simple physics engine, where Rigid Body physics is the main focus, and learn about intersection algorithms for primitive shapes. You’ll begin by building a strong foundation in mathematics that will be used throughout the book. We’ll guide you through implementing 2D and 3D primitives and show you how to perform effective collision tests for them. We then pivot to one of the harder areas of game development—collision detection and resolution. Further on, you will learn what a Physics engine is, how to set up a game window, and how to implement rendering. We’ll explore advanced physics topics such as constraint solving. You’ll also find out how to implement a rudimentary physics engine, which you can use to build an Angry Birds type of game or a more advanced game. By the end of the book, you will have implemented all primitive and some advanced collision tests, and you will be able to read on geometry and linear Algebra formulas to take forward to your own games!
Game Physics Cookbook
Credits
Acknowledgements
Acknowledgements
www.PacktPub.com
Customer Feedback
Preface
Free Chapter
Vectors
Matrices
Matrix Transformations
2D Primitive Shapes
2D Collisions
2D Optimizations
3D Primitive Shapes
3D Point Tests
3D Shape Intersections
3D Line Intersections
Triangles and Meshes
Models and Scenes
Camera and Frustum
Constraint Solving
Manifolds and Impulses
Springs and Joints
Index

## Dot product

The dot product, sometimes referred to as scalar product or inner product between two vectors, returns a scalar value. It's written as a dot between two vectors, . The formula for the dot product is defined as follows:

The sigma symbol means sum (add) everything up that follows. The number on top of the sigma is the upper limit; the variable on the bottom is the lower limit. If n and i is 0, the subscripts 0, 1, and 2 are processed. Without using the sigma symbol, the preceding equation would look like this:

The resulting scalar represents the directional relation of the vectors. That is, represents how much is pointing in the direction of . Using the dot product we can tell if two vectors are pointing in the same direction or not following these rules:

• If the dot product is positive, the vectors are pointing in the same direction

• If the dot product is negative, the vectors point in opposing directions

• If the dot product is 0, the vectors are perpendicular

### How to do it…

Follow these steps to implement the dot product for two and three dimensional vectors:

1. Add the declaration for the dot product to `vectors.h`:

```float Dot(const vec2& l, const vec2& r);
float Dot(const vec3& l, const vec3& r);```
2. Add the implementation for the dot product to `vector.cpp`:

```float Dot(const vec2& l, const vec2& r) {
return l.x * r.x + l.y * r.y;
}

float Dot(const vec3& l, const vec3& r) {
return l.x * r.x + l.y * r.y + l.z * r.z;
}```

### How it works…

Given the formula and the code for the dot product, let's see an example of what we could use it for. Assume we have a spaceship S. We know its forward vector, and a vector that points to its right, :

We also have an enemy ship E, and a vector that points from our ship S to the enemy ship E, vector :

How can we tell if the the ship S needs to turn left or right to face the enemy ship E?

We need to take the dot product of and . If the result of the dot product is positive, the ship needs to turn right. If the result of the dot product is negative, the ship needs to turn to the left. If the result of the dot product is 0, the ship does not need to turn.

### There's more…

Our definition of the dot product is fairly abstract. We know that the dot product gives us some information as to the angle between the two vectors, and . We can use the dot product to find the exact angle between these two vectors. The key to this is an alternate definition of the dot product.

#### Geometric definition

Given the vectors and , the geometric definition of the dot product is the length of multiplied by the length of multiplied by the cosine of the angle between them:

The || operator in the above equation means length and will be covered in the next section. We will cover the geometric definition and other properties of the dot product later in this chapter.