Vectors

I like vectors. It is a very good method to visualize the 3-dimentional world of maths. In algebra we learn all about its properties, the dot- and cross product, matrices etc. But where will I ever use vectors? Physics is one field in which it is commonly used to describe such thing as motion, forces, direction, etc.

Say for example that you have two forces acting on an object on the same point, but now you want to know what the resulting force will be, i.e. where will this object go and with what force or speed. Of course there are ways to calculate this, but why not make it fun? A very easy way to do this is to use vectors, where the length of the vectors will represent the magnitude of the force and the direction of the vectors, the direction of the force. Then you use a bit of geometry to create a rectangle from these vectors, the diagonal of the rectangle will then be the answer you are looking for. Clever isn’t it? This resulting vector is sometimes called the sum of the forces. There are many other applications for vectors, and although it isn’t always necessary to use them, it can make it easier by visualising the problem.

A vector space is a space where the elements are vectors. These spaces have nice properties and are used in many applications. It has a lot of theory behind it and was used in some cases to develop some very deep mathematics. Ever heard of a Hilbert space? It is just a complete vector space of which the norm is defined. Norms are defined differently depending on the space. I will discuss this some other time.

So for now, imagine you are a vector in your space (world) and think about what your direction is (where are you heading?) and what is your length (how great a impact have you made in this world?).