Vectors

The fundamental building block of linear systems is the humble Vector

There is a few different ways to think about vectors. It is not quite right to call them a point, because vectors have a direction that is actually computable. But then they are also not just a pure direction with no associated point.

Personally, I find it best to think of them as a recipe to get to a point based on our understanding of dimensions above. When you plot the vector, it shows you the fastest way of getting to that point in space, which is incidentally, a straight line.

For instance, the vector above is at the position $(2, 2, 2)$ . Another way to think about it might be that it is $2$ steps in the $x$ direction, $2$ steps in the $y$ and $2$ steps in the $z$ direction.

As a slight notational detour, we represent vectors using a kind of shorthand that takes away the $x$, $y$ , $z$ etc and just replaces them with a series of numbers in vertical square brackets, each slot representing a different dimension:

This makes more sense if you look at the most degenerate case and then build up. For instance, take a one-dimensional space, the number line, where we only have an $x$ axis.

Now let us have a look at a vector which ranges around a circle on the$x$ and $y$ axis. Notice that we are still doing the same thing along the x axis, but we are sort of translating the whole line up and down whilst the vector move along the same line.

Extending this to the third dimension is fairly straightforward. If we can imagine our 2D animation running on a flat surface, then in 3D all we are really doing is moving the plane which is that flat surface, around.

Addition and subtraction on vectors is defined in the usual sense. Analytically we just add each component and create a new vector.

Geometrically you can think of this as adding head to tail, or following the steps indicated by the first vector, then following the steps indicated by the second

Of course, note that this only works if the two vectors have the same number of dimensions.

Also note that vector multiplication is not defined in the usual sense - you cannot just take the components of each vector and multiply them together. If you did that, you would lose directional information, because what you are really doing in that case is scaling each component by a different number.