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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).
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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.
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Diagram 5. Vector multiplication
Normalising
A normalised vector or a unit vector has had its magnitude set to one, with the direction left unchanged. A unit vector shows us a direction and can be easily scaled by any value.
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You can find the magnitude (the length) using Pythagorean theorem. A magnitude and unit vector can easily be recombined to create
c = (4, 5)
sqrt(4^2 + 5^2))
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