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Going with the Flow:
A
Vector’s
Tale
Erik Scott Highline CC
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What is a vector?
5,2,3 v
Here are a few examples:
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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?
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Mathematicians and scientists aren’t the only ones who’ve
recognized this fact.
Artists are keenly aware of this, too.
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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
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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
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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
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A vector what?
A “vector field.”
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Van Gogh seemed to find the concept quite natural.
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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.)
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A mathematical representation of our vector field.
Website:
http://math.la.asu.edu/~kawski/vfa2/index.html
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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.)
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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
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One view of a spring’s motion:
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A second interpretation:(units have been adjusted for
simplification)
22
2
1
2
1xvE
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And this can become as complex as you are prepared to handle:
MEpXn
iip
ip
1
TM :