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The Art and Science of Vectors: A Dynamic Exploration

Join Erik Scott on a captivating journey through the world of vectors, from mathematical basics to artistic interpretations. Discover the essence of vectors through real-life scenarios and hands-on activities. Learn how to add, analyze, and understand vector fields with interactive examples. Explore the connection between vector fields, differential equations, and calculus in a visually engaging and informative manner. Website: http://math.la.asu.edu/~kawski/vfa2/index.html

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The Art and Science of Vectors: A Dynamic Exploration

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  1. Going with the Flow: A Vector’s Tale Erik Scott Highline CC

  2. What is a vector? Here are a few examples:

  3. Why an arrow? Compare: What does your eye do with each of the objects below? An arrow is the simplest stationary visual element we can use to convey motion in a specific direction.

  4. Mathematicians and scientists aren’t the only ones who’ve recognized this fact. Artists are keenly aware of this, too.

  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

  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 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

  8. A vector what? A “vector field.”

  9. Van Gogh seemed to find the concept quite natural.

  10. 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.)

  11. A mathematical representation of our vector field. Website: http://math.la.asu.edu/~kawski/vfa2/index.html

  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.”

  14. One view of a spring’s motion:

  15. A second interpretation:(units have been adjusted for simplification)

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

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