Going with the Flow:

A

Vector’s

Tale

Erik Scott Highline CC

What is a vector?

5,2,3 v

Here are a few examples:

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?

Why an arrow?

Mathematicians and scientists aren’t the only ones who’ve

recognized this fact.

Artists are keenly aware of this, too.

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?

4 m/s3 m/s

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.

4 m/s

3 m/s

5 m/s

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

A vector what?

A “vector field.”

Van Gogh seemed to find the concept quite natural.

An activity for the kinesthetic learner.Also known as:

“Pictures are great, but why should our eyes have all the fun?”

1) Stand up. (You are now a simple point.)

2) Point your left arm out towards a neighbor to your left. (Ta-da! You’ve been promoted to a vector.)

3) 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.)

A mathematical representation of our vector field.

Website:

http://math.la.asu.edu/~kawski/vfa2/index.html

Where do vector fields come from?

1. Repeated measurements in many locations. (Like checking currents at different places in a river.)

2. A theoretical understanding of how things change. (Building equations based on an understanding of the forces at work.)

Describing how things change:the domain of Calculus

Vector fields are intimately connected to the mathematical objects called

“differential equations.”

kxdt

xd

2

2

tPcdt

tPd

One view of a spring’s motion:

A second interpretation:(units have been adjusted for

simplification)

22

2

1

2

1xvE

And this can become as complex as you are prepared to handle:

MEpXn

iip

ip

1

TM :