1. Vectors

What is a vector?

• Vector is a mathematical thingy with a length and a direction
• Often represented by an arrow (see next slide)
• Can have multiple dimensions (also called components)
  • In our case usually 2 or 3:
  • , , and possibly
• Video games are full of vectors!
  • Used for depicting position, velocity, acceleration, forces....

2D vector example

• This is a 2D vector

  • x-component
  • y-component

• Vectors start from the origin, or

• In C#:

  Vector2 vectorA = new Vector2(6,3);
  

2D vector length

• The length of a 2D vector is given by the Pythagoras theorem

• In C#:

  float length = Mathf.Sqrt(Mathf.Pow(vectorA.x, 2) + Mathf.Pow(vectorA.y, 2));
  

  • Vector classes have a shorthand, too: A.magnitude

Vector arithmetic

• Manual: Understanding Vector Arithmetic
• Let's introduce the most important vector operations
  • Addition
  • Subtraction
  • Scalar multiplication
  • Extra: vector multiplication
    • Dot product
    • Cross product (WIP)
• C# examples included

Vector addition

• sum of two vectors is calculated by summing up the individual components

• 
  
  

• can be illustrated by moving to start from the endpoint of

  Vector2 A = new Vector2(3.0f, 3.0f);
  Vector2 B = new Vector2(6.0f,-2.0f);
  Vector2 C = A + B;
  

Vector subtraction

• difference of two vectors,

  • is is "flipped" :
  • 
    
    
    
    

• starts from the endpoint of and ends in the endpoint of

  Vector2 A = new Vector2(3.0f, 3.0f);
  Vector2 B = new Vector2(6.0f,-2.0f);
  Vector2 C = A - B;
  

Scalar multiplication

• When a vector is multiplied by a scalar (a number), the vector is scaled

  • If a vector is multiplied by 2, its length doubles
    
    
    
    

  Vector2 A = new Vector2(3.0f,3.0f);
  Vector2 C = 2 * A;
  

Special cases for scalar multiplication

• If the scalar is negative, the vector gets flipped

  • that's what happened in subtraction

• What about division?

  • it's basically multiplication as well

  

Vectors in Unity

Vectors in video games

• position
• velocity
• acceleration
• rotation
• forces

Vector classes

• Unity has two Vector classes, Vector2 and Vector3

  • "2" and "3" here are the number of dimensions: (x,y) and (x,y,z)

  • Vector2 position = new Vector2(1.0f, 2.0f)
    

• Vector components can be accessed with the dot notation:

  • position.x, position.y, position.z

• Vectors can't be directly modified:

  • No: position.x = 3.0f

  • position = new Vector2(3.0f, position.y);
    

• Length of a vector can be acquired with vector.magnitude

  • Script Reference: Vector3 magnitude

Velocity vector

• Velocity is an important concept in game development!
• It's the rate of change of position
  • new_position = old_position + velocity;
• 3-dimensional velocity vector can be added like this to GameObject's position in an Update() method:
  • transform.position += velocity;
  • We can use player input, collision, etc. to adjust the velocity as needed

Note: Velocity vector is usually drawn so it starts from the moving object

• (Remember, though, that vectors do not "know" its starting positions.)

  

Acceleration vector

• Another important concept is acceleration
• It's the rate of change of a GameObject's velocity!
  • new_velocity = old_velocity + acceleration;
• We can use acceleration to change Rigidbody's velocity
  • rb.velocity += acceleration;
• We can't use that for GameObjects without rigidbodies, but we can store velocity in a separate variable

  velocity += acceleration;
  transform.position += velocity;
  

• This might seem like a cheap trick, but actually allows us to create smoother motion if we use player input, collision, etc to control acceleration instead of velocity

Distance vector

• To get the distance vector between two objects, we use vector subtraction
  • vector_B - vector_A
  • transform.position - otherGameObject.transform.position;
• To just get the length of the distance vector, a.k.a, the distance:
  • Vector2.Distance(vector_A, vector_B)

Special vectors of Unity

• See Static properties in Script Reference: Vector3
  • Vector3.right: the global axis
  • Vector3.up: the global axis
  • Vector3.forward: the global axis

Common vector operations

Normalizing a vector

• When we are not interested about the length of a vector, only its direction, it helps to normalize the vector
• Normalizing means setting the length of a vector to be
• This is achieved by dividing the vector by its length
• in C#, the normalized version of any vector can be easily accessed:

  Vector2 UnitVector = A.normalized;
  

Rotating a vector

• In Unity, rotation is represented by Quaternions (fin. kvaternio, kvaterniot)
• To rotate a vector by a given angle, you can do a Quaternion rotation operation:

  Vector3 rotatedVector = Quaternion.Euler(0, 0, 90) * originalVector;
  

  • This isn't a regular multiplication, so do note that Quaternion.Euler has to be on the left side of the vector.
• To rotate a Quaternion:

  myQuaternion *= Quaternion.Euler(0, 0, 90);
  

• Note: the Transform and Quaternion classes have many rotation methods available, see Transform Class: Rotation

Extra: Dot product - Vector's alignment with another vector

• If you want to know how much two vectors point to the same direction, we can use the dot product
  • Script Reference: Vector3.Dot
• The dot product returns a number, not a vector!
• For normalized vectors, Vector3.Dot returns
  • if they point in exactly the same direction
  • if they point in completely opposite directions
  • if the vectors are perpendicular

Dot product example

• Check if two rigidbodies are moving to the same direction:

  float alignment = Vector3.Dot(
    _rigidBody.velocity.normalized,
    otherGameObjectRigidBody.velocity.normalized)
  

Note about distance

• Note: when performance is important, using magnitude can be a bad idea: it includes the costly square root operation
  • if you need to only compare magnitudes, or you're squaring it right away, use .sqrMagnitude instead!

Exercise 1. I'm being avoided

Create a scene with two GameObjects, a player and a static enemy.
Calculate the distance between the player and the enemy.

If the distance is smaller than some given threshold value,

• Change the color of the enemy (and set the color back to default when you're no more on the range)
• Make the enemy shoot at player!
• Move the enemy farther away from the player along the shortest possible path.

Reading & watching

• Manual: Vector Cookbook

• Brackeys video: Vectors

• Freya Holmér video: Vectors & Dot Product