1. Going with the Flow: A Vector’s Tale Erik Scott Highline CC
Presented by Erik M. Scott at the Fall 2001 Science Seminar.
Last spring I gave a talk on the notion of dimension. I had a good idea for the talk well in advance, worked with some math graduate students on technical details for a month, and gave what turned out to be a pretty good talk. By the end of spring, Eric had asked if I would talk in the fall. I said yes, figuring that I’d be able to get the talk together over the course of a summer and half a quarter. Before long, I even had a clever title. I thought it was great because it contained puns on certain mathematical ideas I’d present.
Little did I know that it also contained a reminder for me and my audience related to the execution of my talk: go with the flow. While I’d describe my talk in spring as having a laminar, or very smooth and direct flow, this one will probably be a bit more turbulent or chaotic. So as I say, “Go with the flow.”
Erik Scott
Highline CC

2. Here are a few examples:
What is a vector?
Here are a few examples:
Let’s start with the basics: What is a vector?
Webster defines a vector as: 1. An organism that transmits disease, or 2. A quantity that has magnitude and direction.
To these I add 3. A car manufacturer.
What do the images have in common? They all reflect motion in some way. And that’s the essence of a vector.

3. Compare: What does your eye do with each of the objects below?
Why an arrow?
An arrow is the simplest stationary visual element we can use to convey motion in a specific direction.
Compare: What does your eye do with each of the objects below?
For me, when looking at the dot my eye just sits there, kind of like people at a typical math or science lecture. On the other hand, the arrow causes my eye to move, more like the audience at presentations by Derek Greenfield. So much so, in fact, that I keep feeling like the yellow arrow should be pointing AT something, or is suggesting there is something profound on the next slide – it’s very irritating.

4. Mathematicians and scientists aren’t the only ones who’ve recognized this fact.
Artists are keenly aware of this, too.
The Powerpuff Girls show something of an arrow shape with their bodies. The waterfall doesn’t necessarily use arrows, but the long strings of water draw our eyes in a particular direction just like the arrow. Our concept of gravity and things falling down does the rest.

5. My (formal) introduction to vectors:
A river flows south at four meters per second, and a person wants to swim across. The person tries to swim straight ahead at three meters per second. What is the person’s actual heading?
3 m/s
4 m/s
This was one of the first problems I had to solve with vectors. It taught me that you can use algebra and geometry when working with vectors.

6. Solution idea:
Add vectors head-to-tail, then draw a final arrow connecting the tail of the first vector with the head of the last. That’s your direction. Calculations give you the speed.
3 m/s
4 m/s
(Explain)
5 m/s

7. Important features of the example:
In this situation, everything moves at a constant speed. That’s what allows us to use only algebra and plane geometry.
4 m/s
In this picture, you can see that the arrows are all moving in the same direction and are of the same length. This has the fancy name of “uniform laminar flow.”
The picture also gives you an example of what we call a vector field.

8. A vector what? A “vector field.”
In the last picture, we had vectors located at various points in the river to show the current, or motion of the water. That abstraction, of assigning a vector or arrow to each point in space, creates what is known as a vector field. You can even move around the picture just by looking at individual filaments. It’s as if your eye “flows” along a path. And we use the word exactly that way: your eye traces “flow lines.”

9. Van Gogh seemed to find the concept quite natural.
Here you see the same thing.

10. An activity for the kinesthetic learner
An activity for the kinesthetic learner. Also known as: “Pictures are great, but why should our eyes have all the fun?”
Stand up. (You are now a simple point.)
Point your left arm out towards a neighbor to your left. (Ta-da! You’ve been promoted to a vector.)
Take the paper ball with your right hand and pass it on with your left. (You’ve just become part of a vector field and created a flow line.)
Vector field of people.

11. A mathematical representation of our vector field.
Website:

12. Where do vector fields come from?
Repeated measurements in many locations. (Like checking currents at different places in a river.)
A theoretical understanding of how things change. (Building equations based on an understanding of the forces at work.)

13. Describing how things change: the domain of Calculus
Vector fields are intimately connected to the mathematical objects called
“differential equations.”
Upper eqn: spring
Lower eqn: population growth

14. One view of a spring’s motion:
Orient them. Talk about the potential problem of smearing.

15. A second interpretation: (units have been adjusted for simplification)
Show vector field on website.
Wanted to do other but freakin’ negative got in the way.

16. And this can become as complex as you are prepared to handle:
Math is hard.
Show sensitivity to initial conditions.