Vector: Basics. Products. Spaces. Matrices. - Gr8-Tools/game-developer-roadmap-doc GitHub Wiki

Total time: 16.5 hours

Vector: Introduction

In the world of game development, vectors play a crucial role in various aspects such as physics simulations, graphics rendering, and character movement. Understanding the basics of vectors is essential for any aspiring game developer. In this article, we will explore the fundamentals of vectors and their applications in game development.

What is a Vector?

A vector is a mathematical entity that represents both magnitude and direction. It is commonly represented as an arrow, where the length of the arrow represents the magnitude, and the direction of the arrow represents the direction of the vector. In game development, vectors are typically used to represent positions, velocities, forces, and orientations.

Components of a Vector

A vector can be broken down into its components, which are the individual values that contribute to its overall magnitude and direction. In a two-dimensional space, a vector has two components: the x-component and the y-component. In a three-dimensional space, a vector has three components: the x-component, the y-component, and the z-component.

Vector Operations

There are several operations that can be performed on vectors, which are essential for manipulating and working with them in game development. Some of the common vector operations include:

1. Addition: Two vectors can be added together by adding their corresponding components. For example, if we have two vectors A and B, the sum of A and B is obtained by adding their x-components and y-components separately.

2. Subtraction: Similar to addition, subtraction of vectors involves subtracting their corresponding components.

3. Scalar Multiplication: A vector can be multiplied by a scalar value, which scales the magnitude of the vector without changing its direction. This operation is useful for adjusting the intensity of forces or velocities.

4. Dot Product: The dot product of two vectors yields a scalar value. It is calculated by multiplying the corresponding components of the vectors and summing them up. The dot product is useful for determining the angle between two vectors or projecting one vector onto another.

5. Cross Product: The cross product of two vectors yields a new vector that is perpendicular to both input vectors. It is calculated by taking the determinant of a matrix formed by the input vectors. The cross product is commonly used in physics simulations and calculating surface normals.

Applications in Game Development

Vectors find numerous applications in game development. Here are some of the areas where vectors are commonly used:

1. Physics Simulations: Vectors are used to represent forces, velocities, and accelerations in physics simulations. They are crucial for realistic movement and interactions between objects in the game world.

2. Graphics Rendering: Vectors are used to represent positions and orientations of objects in 3D graphics rendering. They are essential for transforming and manipulating objects in the virtual space.

3. Character Movement: Vectors are used to control the movement of characters in games. By applying forces and velocities to characters, their positions and animations can be updated accordingly.

4. Collision Detection: Vectors are used to detect and handle collisions between objects in games. By comparing the positions and sizes of objects, vectors can be used to determine if they intersect or overlap.

In conclusion, understanding the basics of vectors is crucial for game developers. Vectors are versatile mathematical entities that find applications in various aspects of game development, including physics simulations, graphics rendering, and character movement. By mastering vector operations and their applications, game developers can create more immersive and realistic gaming experiences.

Additional materials

Articles

1. Math is Fun - Vectors. 30 minutes.
2. Unity Vector3 API Documentation: The Vector3 API documentation from Unity provides detailed information about the Unity's built-in Vector3 class and its methods. Reading time: Variable, depending on the specific methods explored.

Videos

1. Vector & Dot Products: Introduction to vectors world. Here explained basics of vector operations, including Dot products.

7.5 hours

2. Spaces & Cross products: Continuation of the video above. Here we are diving deeper in vector operations (Cross Products), gettings acquainted with vector spaces and matrix representation of vector and it's operations.

6 hours

3. Linear transformations with Matrices: Explanation of mapping vectors in different spaces with help of Matrices.

30 minutes

Tasks

Math Lib

The following tasks should be realised in new project "Math Lib". Write functions that resolve the tasks issues in one class and demonstrate the functionality in any way on "Scene"- or "Editor"-windows in another one. We will use this project to create our own implementations of this Maths functions.

1. "Vector Operations": Write a program that performs vector addition, subtraction, and scalar multiplication. The program should take two vectors as input and output the result of each operation.

30 minutes

2. "Vector Normalization": Write a program that takes a vector as input and normalizes it to have a magnitude of 1. The program should output the normalized vector.

15 minutes.

3. "Vector Dot Product": Write a program that calculates the dot product of two vectors. The program should take two vectors as input and output the dot product result.

15 minutes.

4. "Vector Cross Product": Write a program that calculates the cross product of two vectors. The program should take two vectors as input and output the cross product result.

15 minutes.

5. "Space transformations": Write a program that represent vector from one space to another. The program should take two normal-vectors that set up spaces ("source" and "target") and one more vector in "source" space. Program should output this vector with coordinates of "target" space.

Use matrices.

1 hour.