Lecture 9: Vectors

# Creación de Videojuegos

### Vectors

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# Math

Cady: *I like math.*

Damian: *Eww. Why?*

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

*Mean Girls (2004)*

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# 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

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# Vectors

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# 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
   
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# Vectors

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

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

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

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# Vector Operations

* A vector describes a *change* in position
  
  * As such, a vector does not itself have a position 
  
  * That means, the vector that moves a point `a` by 3 units in x direction and 2 units in y direction is the same as the vector that moves a point `b` by 3 units in x direction and 2 units in y direction 
  
$$
\begin{pmatrix} 
3 \\\\ 
2
\end{pmatrix}
$$
  
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# Vector Operations
  
  * For two points a and b, b - a gives you a vector: the change required to go **from a to b**
  
  * `$t \cdot (b-a)$` gives you a "part" of that change
  
  * `$a + t \cdot (b-a) $` gives you a point on the line from a to b. t determines where on the line that point is:
     - 0: The point is a 
     - 0.5: The point is halfway between a and b 
     - 1: The point is b 
     - greater than 1: The point is on the opposite side of b than a
     - etc.
     
  * You can interpret `t` as a "percentage"
  
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# 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?

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# 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!

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# 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.

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# 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

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

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# 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?

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# 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?
  
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# 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)
  
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# Sin and Cos

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# Projection

What if B is a unit vector?
  
$$
\vec{A} \cdot \vec{B} = \cos(\alpha) \cdot |\vec{A}|
$$
  
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# Dot Product: Projection

If B is a unit vector:

$$
\vec{A} \cdot \vec{B} = \cos(\alpha) \cdot |\vec{A}|
$$
  
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# Dot Product: Example 1

* An AI controlled enemy is at `$\begin{pmatrix} 1 \\\\ 1\end{pmatrix} $`
  
  * They are currently looking in *direction* `$\begin{pmatrix} 2 \\\\ 0\end{pmatrix} $`
  
  * The player is at `$\begin{pmatrix} 5 \\\\ 4\end{pmatrix} $`
  
  * The enemy has a field of view of 45 degrees in each direction. Can it see the player? (`$\cos(45) = \frac{\sqrt{2}}{2} \approx 0.7 $`)  
  
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# Dot Product in 3 Dimensions

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

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# Next Time: 3D Graphics

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# References

* [Linear Algebra Video Series](https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)

* [Understanding the Dot Product](https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/)
 
 * [Matrix-Vector Multiplication](https://mathinsight.org/matrix_vector_multiplication)