Math

Cady: I like math.

Damian: Eww. Why?

Cady: Because it's the same in every country.

Math

Cady: I like math.

Damian: Eww. Why?

Cady: Because it's the same in every country.

Mean Girls (2004)

Math

• In video games we need a lot of math

• In particular, we need a lot of vector math

• Generally, game objects will be located somewhere, move, collide, etc.

• Today we are going to revisit how vectors work

Vectors

• How do we represent positions, movements, etc.? Vectors!

• A vector has 2 (or 3) components, called x and y (and z)

• Addition and subtraction work component-wise

• You can also multiply a vector with a real number

Vectors

$$\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} x+a \\ y+b \end{pmatrix}$$

$$a \cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a \cdot x \\ a \cdot y \end{pmatrix}$$

$$\begin{pmatrix} x \\ y \end{pmatrix} - \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} x-a \\ y-b \end{pmatrix}$$

Example

• The player is currently at $(7,10)$

• The enemy is at $(1,2)$

• Which direction does the enemy have to walk in, to get to the player?

• What point is a quarter (0.25) of the way from the enemy to the player?

Velocity?

• Now we know the "direction" the enemy has to walk in

• But usually we want our game objects to move with constant velocity

• How do we ensure that?

Velocity?

• Now we know the "direction" the enemy has to walk in

• But usually we want our game objects to move with constant velocity

• How do we ensure that?

• Unit vectors!

The length of a Vector

$$\left| \begin{pmatrix} x \\ y \end{pmatrix} \right| = \sqrt{x^2 + y^2}$$

$$\left| \begin{pmatrix} x \\ y \\ z \end{pmatrix} \right| = \sqrt{x^2 + y^2 + z^2}$$

A unit vector is a vector with length 1.

Normalizing a Vector

• A unit vector is useful to give us a general direction

• We can later multiply it with the desired distance

• To turn a vector into a unit vector, we divide it by its length

$$\vec{v}_n = \frac{\vec{v}}{|\vec{v}|}$$

Example

• The player is currently at $(7,10)$

• The enemy is at $(1,2)$

• The enemy is moving with 2 units/s towards the player

• Where is the enemy after 2 seconds?

• When will the enemy reach the point that is 80% of the way to the player?

Vector Rotation

• To turn a vector by 90 degrees: Exchange the coordinates and switch the sign of one, i.e. (x,y) becomes (-y,x)

• How could we return by other angles?

• We'll talk about that next week

• But maybe we could start by calculating the angle between two vectors?

Vector Operations: The Dot Product

$$\vec{A} \cdot \vec{B} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} \cdot \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = x_1 \cdot x_2 + y_1 \cdot y_2\\ \vec{A} \cdot \vec{B} = \cos(\alpha) \cdot |\vec{A}| \cdot |\vec{B}|$$

• Very useful for a variety of applications!

• Simplest: Calculate the angle between two vectors.

• Calculate the projection of a point onto a vector (e.g. distance from a line)

Sin and Cos

Projection

$$\vec{A} \cdot \vec{B} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} \cdot \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = x_1 \cdot x_2 + y_1 \cdot y_2\\ \vec{A} \cdot \vec{B} = \cos(\alpha) \cdot |\vec{A}| \cdot |\vec{B}|$$

What if B is a unit vector?

$$\vec{A} \cdot \vec{B} = \cos(\alpha) \cdot |\vec{A}|$$

Dot Product: Projection

If B is a unit vector:

$$\vec{A} \cdot \vec{B} = \cos(\alpha) \cdot |\vec{A}|$$

Dot Product in 3 Dimensions

$$\vec{A} \cdot \vec{B} = \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} \cdot \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} = x_1 \cdot x_2 + y_1 \cdot y_2 + z_1 \cdot z_2\\ \vec{A} \cdot \vec{B} = \cos(\alpha) \cdot |\vec{A}| \cdot |\vec{B}|$$

References

• Linear Algebra Video Series

• Understanding the Dot Product

• Matrix-Vector Multiplication