Vectors

Vectors… Let’s talk about them. There will be almost no math here, we have that elsewhere.

Our interest is almost entirely Physics. We argue that we chose the math we teach you because we discovered that it could explain our physical data. This is true at least for ‘displacement’, ‘viscosity’, ‘acceleration’, ‘force’, ‘momentum’, ‘energy’, ‘power’.

The vectors we use are sometimes called geometric vectors. It’s a physics thing. We accept that math people are free to push thinking in all kinds of directions and that they are adhering to the adage “follow the truth where it leads you”. Sometimes math people do great work when the only rule on the table is “everything you put together must be self-consistent”.

When the gods of math created Vector Space they may have built it thinking exclusively about geometric vectors. With that in mind, try the following story aboud cardinals and birds:

Someone who loves cardinals builds houses for cardinals, but the houses they built were soon occupied by several other bird species (blue jays, cedar wax wings) and it was decided later that anything that could live in the bird-house must be a bird.

We extend the above analogy and say that geometric vectors are cardinals and that the word vector as used by a mathematician applies to all birds that can live in the houses. What we call Vector Space is analogous to the bird house. One problem with this analogy is that the math people were not willing to look for a word more general than “vector”. If they had they could have used that to replace the word ‘Vector’ in Vector Space.

You can read in “Advanced Calculus” by R. Creighton Buck that we can add points. Creighton’s textbook on Advanced Calculus is a beautiful read. He talks to you about a difficult topic, and it makes sense! Creighton is a valuable dissident doing work that is appropriate in places outside of where we do Tensor Calculus.

This leaves us with the prerogative of defining a subset of what we will allow ourselves to do. Armed guards at the various gates leading to Vectorville will turn away lists of numbers and things line points and functions. Go to Mathworld! (and it isn’t meant to be mean)

Our geometric vectors have direction and magnitude. Each of these two notions deserves its own blog. We should add that magnitude can be broken into a numerical component and unit of measurement. A prof might tell you that we get this by adding structure to the base definition of a vector (presumably the one that math people tell you is the most basic one or they might say it is the simplest thing that can tell the world “I am a vector!”)

We might tell you that a vector exists by being a something that has an initial point and a final point. From this description we can pull two stories. In one story the vector just sits there with the initial point and the final point. This is probably the better of the two stories but you will have use for the other. In the other story we start with the initial point and the vector is the “action” or “motion” or “whatever” that moves the point to the location where it is the final point. In this second story we begin with a point and we end with a point. We might even argue that the vector is destroyed when it does what it wanted to do.

If you like that you might also like another “try to see a vector in different ways” story found in “Tensor Calculus” (a Schaum’s Outlines book) where David Kay discusses “alias” and “alibi”. For this discussion assume that we first described a point as (4,3) and then later we described it as (5,0).

alias – The point exists in two different coordinate systems and the different numbers come about because of differences in the coordinate systems.

alibi – The point moves from a first location (4,3) to the second location (5,0) and both locations are in the same coordinate system.

We run into the notion described above when we use a matrix to change a first vector to a second vector. We can use a matrix that provides rotation. The math person will tell you that what the matrix does could be viewed as the vector holding still and the coordinate system rotating or it could be viewed as the coordinate system holding still and the vector rotating. The math person might even stare at you and say “this is showing you the relativity that is important to you.”

Elsewhere, in a Tensor Calculus class taught by a physics professor, you are quietly told that we respect “invariance” and therefore the vector doesn’t move, the rotation has to be a rotation of the coordinate system.

Bonus Features

“Paper/Screen”

Chances are almost all your work with vectors happens on a piece of paper or some sort of computer screen. There may be times it seems to you that the paper or the screen is a part of the tale you tell. We just want you to know others have said the same thing. This may be more likely when you are doing some work with drawn arrow vectors and there is no coordinate system.

“Independent Vectors”

For part of the tale we tell you that vectors can exist on paper or screen independent of a coordinate system. The way this is told helps you and you might jest later that it’s funny how we do our work almost entirely with vector component numbers even though that requires a coordinate system and we have sworn oaths affirming vector independence.

It might help to change the tale a bit and say that we probably draw those independent vectors with a coordinate system taped behind the glass so we can get the scaling of all vectors correct, and then we remove the coordinate system and tell you what we have drawn is independent.

One prof explained it as “yes, there is a coordinate system–we just don’t have to tell you which one”.