A vector is a magnitude and a direction carved in perfect glass.
A vector tells Mother Nature: “I know what I am, what I am is my direction and my magnitude and nothing more.”
Mother Nature says to the vector: “I know, I will carve you with gold in perfect glass and you will never change.”
We can hold a transparency with the lines of coordinates on it behind the glass and take a measurement using our coordinate transparency and write down numbers. But Mother Nature will flash lightning and in our fright we drop the transparency and the paper and the numbers are gone–but the vector remains, a magnitude and a direction and nothing more.
We can cooperate with Mother Nature once we are smart enough to stop fighting her. We can hold up our transparency to the glass and using two points from the transparency, we ask Mother Nature to draw us a vector.
- Drop the transparency, let it fall to the floor.
- What is drawn in perfect glass is a magnitude and a direction and nothing more.
A point is just a list of numbers. And it is just one point.
It might tell us, “I can have a second point, I will grab the origin. Draw an arrow to me from the origin!”
In so saying, it becomes something different. It has turned itself into a horror, and every time we hold up a different coordinate system the drawn arrow changes because it must be that list of numbers connected to the origin.
And it screams, because change causes pain.
Let a point stay a point. Let a vector be a vector.
Appendix A
If you actually do this you will have to have an origin on your coordinate system transparency and you have to match that origin to the tail of the vector.
Furthermore, you will need to have instructions on the transparency explaining how to orient it when you hold it up to the glass. Perhaps they will create a standard some day where if there is a certain word on the transparency you know align the top of the transparency with the top of the glass.
Appendix B
There is something beautiful about the work we do with numbers. What Descartes developed was incredible!
Descartes developed something that interferes with our ability to catch several notions that we need for the understanding of tensors. It gets in our way, like a grinning cat. We love that cat.
Appendix C
One instructor remarked that in a way we create a problem–what we have is as much about the coordinate system as it is about the vector.
Vectoreverything 