\( \definecolor{flatred}{RGB}{192, 57, 43} \definecolor{flatblue}{RGB}{41, 128, 185} \definecolor{flatgreen}{RGB}{39, 174, 96} \definecolor{flatpurple}{RGB}{142, 68, 173} \definecolor{flatlightblue}{RGB}{52, 152, 219} \definecolor{flatlightred}{RGB}{231, 76, 60} \definecolor{flatlightgreen}{RGB}{46, 204, 113} \definecolor{flatteal}{RGB}{22, 160, 133} \definecolor{flatpink}{RGB}{243, 104, 224}\)

April 17, 2021

Vectors in the Plane

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Textbook Reference: Section 8.8 Vectors

The story of mathematics is the story of invention.

It's a story you've seen retold time and time again.

Think about that for a second. How many times have you seen the following story play out in your various math classes?

  • First, you learn about some things, and you give those things names.
  • Then you learn how to combine those things in mathy ways to get new things.

When you first started learning, those things were called "numbers", and you combined them in ways called "addition", "subtraction", "multiplication", and "division". As you got to higher and higher grades, you learned about new kinds of numbers these same methods could be applied to, such as negative numbers and fractions.

But you've seen this same story unfold with things like matrices, sets, or even geometric objects. The way that you combine things might change here and there, but the idea is that you're looking for patterns and structure.

That's what we're going to do now as we look at the math of vectors.

What is a vector?

There are two ways we can understand vectors:

  • On one hand, a vector can be thought of as a list of numbers. This is how computer scientists often think of vectors — as a way to hold data.
  • On the other hand, a vector can be thought of as an arrow, having some defined length and pointing in some direction. This is how physicists often think of vectors — arrows designating things like force, velocity, and so on.

This arrow represents the vector \(\color{flatred}\langle 3,4\rangle\).

So now we have three questions we need to answer:

  • How do we do math with lists of numbers?
  • How do we do math with arrows?
  • And how can we reconcile these two seemingly different ways of thinking about vectors?

The beautiful thing here is, in some sense, we get to decide. There are a lot of possible definitions we could make. Some definitions might be more useful than others. But the point is, we as mathematicians are the ones in charge here.

So, in the following Preview Activity, you're going to do some math with vectors, despite not having been told explicitly how to do so. The point is for you, the mathematician, to make the choice of what makes the most sense, and piece the puzzle together.

Preview Activity 27

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

  1. First let's consider the first way of thinking of vectors: as lists of numbers.

    Answer each of the following questions, based only on your own intuition. In other words, if you were the one in charge of the definitions, give the answer that you think would make the most sense. Explain the choices you've made.

    1. \({\color{flatred}\langle 3,4\rangle}+\color{flatblue}\langle -2,5\rangle\)
    2. \(2\color{flatred}\langle 3,4\rangle\)
    3. \(-\color{flatblue}\langle -2,5\rangle\)
    4. \(2{\color{flatred}\langle 3,4\rangle}-\color{flatblue}\langle -2,5\rangle\)
  2. Now let's consider the second way of thinking of vectors: as arrows.

    In the following picture, the red vector is \(\color{flatred}\langle 3,4\rangle\) while the blue vector is \(\color{flatblue}\langle -2,5\rangle\).

    Here you have the same four calculations as in #1. See if your answers you put above make sense. If not, should there be a different answer that would make more sense?

    1. \({\color{flatred}\langle 3,4\rangle}+\color{flatblue}\langle -2,5\rangle\)
    2. \(2\color{flatred}\langle 3,4\rangle\)
    3. \(-\color{flatblue}\langle -2,5\rangle\)
    4. \(2{\color{flatred}\langle 3,4\rangle}-\color{flatblue}\langle -2,5\rangle\)
  3. Answer AT LEAST one of the following questions:
    1. What was something you found interesting about this activity?
    2. What was an "a-ha" moment you had while doing this activity?
    3. What was the muddiest point of this activity for you?
    4. What question(s) do you have about anything you've done?

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