Vectors are enormously useful geometric object in maths and physics. Vectors express a direction and a magnitude (or length), in both 2D and 3D space. Vectors are key concept not only for producing geometry through code, but also for motion and interactive animation.
Diagram 1. Euclidean vector
The direction of a vector is simply represented with an arrow, with the length of the arrow expressing the magnitude. The information to create a 2D vector can be easily recorded with just two numbers, which can be negative or positive.
a = ( 3, 6 )
Diagram 2. Vector a
The key thing to remember is that vectors represent direction and magnitude without a location. For this reason vectors are often combined with a coordinate. Confusingly, 2D and 3D coordinates are generally recorded in the same format as vectors, with just two numbers. However, coordinates are normally visually represented as points.
Diagram 3. Vector a, translated to point b (5,3)
There are a number of ways to manipulate and combine vectors with simple operations (vector maths).
Addition and Subtraction
If you add to vectors together, the resulting vector is the equivalent of stacking the vectors end to end. Adding vectors together can be used to simulate simple physics like the effect of wind or gravity, so for every step forward a game character might have a wind vector added to their movement vector. If a character in a game jumps while moving forward, then we also want to add the vectors of the upward movement with the forward motion, and then in reverse when they fall back down.
a = ( 3, 6)
b = ( 7, 2)
a + b = (10, 5)
Diagram 4. Vector addition
Scaler Multiplication
Scaler Multiplication only effects the magnitude of the vector, leaving the direction unchanged. However, if the vector is multiplied by a negative, then the direction is reversed! Scaler Multiplication can be used to control the acceleration of space ship ot to simulate wind resistance or drag. If an object in a game collides with wall for example, we may want to multiply the objects vector by -1, to reverse it's direction so that it bounces off the surface.
Diagram 5. Vector multiplication
Diagram 6. Vector negative multiplication
Normalising
A normalised vector has its magnitude set to one, with the direction left unchanged.
Finding the magnitude
You can find the magnitude (the length) using Pythagorean theorem:
(Diagram 7. sqrt(x^2 + y^2))
PVector
Processing has a vector class called ‘PVector‘, which can be two or three dimensions. Basic vector maths are included in a number of methods of the PVector class.
PVector vec1 = new PVector(1,2); PVector vec2 = new PVector(3,4); println(vec1.x); println(vec1.y); println(vec1.z); println(vec1); println(vec2); vec1.set(10,20); println(vec1); println("-- Vec Add --"); PVector resVec = PVector.add(vec1,vec2); println(resVec); vec1.add(vec2); println(vec1); println("-- Vec Sub --"); resVec = PVector.sub(vec1,vec2); println(resVec); vec1.sub(vec2); println(vec1); println("-- Vec Mult --"); resVec = PVector.mult(vec1,2); println(resVec); vec1.mult(2); println(vec1); println("-- Vec Length --"); println(vec1.mag()); println("-- Vec Angle Between --"); println(degrees(PVector.angleBetween(vec1, vec2))); println("-- Vec Normalize --"); vec1.normalize(); println(vec1); println(vec1.mag());