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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 1.9 Representations of Vectors

Exercises Practice Activities

1. Representations of Volume.

You want to determine the volume of the room you are in.
  1. Write a description of how to find the volume of the room in words.
  2. Sketch a diagram that would help you find the volume of the room.
  3. Write a symbolic expression that would allow you to find the volume of the room. Check the units of your expression.
  4. Without standing up or using a calculator, estimate the volume of the room as a number. Make sense of the number by comparing it to something.

2. Metacognition.

Make a list of the kinds of representations you used during the previous activity, and how they were helpful. Explicitly identify any assumptions you made during the activity. Why do you think you were asked to do this activity? What were the advantages of not using a calculator?

3. Summarize Representations.

Write down a list of things about vectors you have learned so far. Organize your thinking by representation: (a) words or sentences, (b) numbers or symbols, and (c) pictures or diagrams.

4. Vectors in the Garden.

You visit a garden with a trail that includes the following landmarks.
  • Red roses at \(\vec{r}_r = -3a\hat{x}\)
  • White roses at \(\vec{r}_w = +4a\hat{x}\)
  • A pond at \(\vec{r}_p = 0\hat{x}\)
  • A bench at \(\vec{r}_b = -a\hat{y}\)
  • A bridge over a creek at \(\vec{r}_c = -a\hat{x}\)
  • A statue at \(\vec{r}_s = +2a\hat{x}\)
  1. Sketch and label the garden and its landmarks.
  2. Find the following displacement vectors using both symbols and diagrams. (a) From the red roses to the white roses, (b) From the pond to the red roses, and (c) From the bench to the statue
  3. Do some physics sensemaking about the situation and about your answers.

5. A Pair of Vectors.

You have two vectors that have the same magnitude \(a\text{,}\) but can point in any direction. Sketch the vectors in each case below. Your sketches should be quantitatively accurate (you should know the angle between the vectors).
  • The magnitude of the sum is \(0\text{.}\)
  • The magnitude of the sum is \(2a\text{.}\)
  • The magnitude of the sum is \(\sqrt{2}a\text{.}\)
  • The magnitude of the sum is \(\sqrt{3}a\text{.}\)
  • The magnitude of the sum is \(b\text{.}\)
Make sense of your answer with the following questions: What is the largest possible sum? What is the smallest?
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