So, what exactly is a ‘vector‘?
Like so many other words, it seems to mean something different in different ‘disciplines’. But, deep down, the meanings are connected through the root of the word, ‘vector‘. In Latin, it means ‘carrier’ – so, in all its uses now, it implies a direction, an increase or decrease, or another dynamic component to the basic information it, well, carries.
Recently, I have read a most excellent – clear and understandable – explanation of what it is that a ‘vector’ is. So, with CodeSlinger’s permission (and a few illustrations and links thrown in by me), without further delay…
VECTORS: a tutorial by CodeSlinger
On the plane, pick any three points, not lying on the same line. Call one of them the origin. Call the line from the origin to the second point a basis vector in the u-direction, and call the line from the origin to the third point a basis vector in the v-direction.
Then you can represent any point on the plane by a sum of appropriately scaled copies these two vectors.
No amount of scaling will turn either of these vectors into the other. Thus we say they form a basis of the plane, which is what were anticipating when we called them basis vectors in the first place.
Of course, this basis is not unique. Any two vectors which are not parallel form a basis of the plane in which they lie.
However, it always takes two of them, so we say the plane is a two-dimensional space.
Similarly, we can find triples of vectors which form bases of three-dimensional space, and N-tuples of vectors that form bases of N-dimensional space.
If a set of vectors forms a basis for a space, then we say that the basis spans the space. The essential defining characteristic of a basis is that none of the N vectors can be obtained by any combination of scaled copies of the remaining N-1 vectors in the set.
Now, if we choose our basis vectors such that the angle between any pair of them is always 90 degrees, then our basis has the additional benefit that the directions are mutually independent. No amount of movement parallel to any basis vector results in any movement parallel to any of the others.
When a basis has this property we call it an orthogonal basis. Going back to our plan , for example, we now have the x and y directions familiar from graphing.
In three-dimensional space, we have x, y and z. And the idea extends to N-dimensional space, even though there may not be standard names for the basis vectors. So, now that we have a clear picture of the properties of basis vectors in geometric terms, let’s get a little more abstract.
We can treat any set of N mutually exclusive and jointly exhaustive properties (also called degrees of freedom) as a set of N basis vectors in an N-dimensional representation space.
A representation space is just the set of all possible combinations of these properties. For example, red, green and blue. Any colour visible to the human eye can be represented as a sum of appropriately scaled red, green and blue components, but neither red, green nor blue can be obtained by any scaled mixture of the other two.
Thus we say that red, green and blue are the 3 basis vectors of a 3-dimensional colour space.