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## The Distance Between Two Points: Instructor's Guide

### Main Ideas

- Multiple representations of a displacement vector
- Geometric understanding of displacement vectors
- Displacement vectors located away from the origin of coordinates

### Students' Task

*Approximate Time: 20-30 minutes*

To figure out how to describe the vector between Captain Kirk and Mr. Spock to Enterprise.

### Prerequisites

Some knowledge of vectors

### Props/Equipment

- More than 3 students
- Blackboard

### Activity: Introduction

We do this activity by having students act out a “Star Trek” activity. Other scenarios are, of course, possible. If you want to do Star Trek, reassure students that might not know about Star Trek that all they need to know is that Kirk, Spock, and Scotty are officers on a starship. The starship has the ability to “beam” people around, i.e. move them from anywhere to anywhere via advanced technology called a transporter that requires lots of energy. This is the way the story goes:

Ask for volunteers to play the roles of Captain Kirk and Mr. Spock. The instructor takes the role of Scotty. (Stand on the table to indicate the you are orbiting in a spaceship.) Everyone else is a “red shirt” who wants to be promoted to a shirt of another color. The “red shirts” are the one-time characters in the show who can be killed off without long-term consequences to the show. To be promoted to a shirt of another color, students will have to impress Scotty with their calculational ability so that he will recommend them.

Kirk and Spock are in separate places in a city on the surface of a new planet. Kirk is under attack by aliens. But the ship is also under attack by aliens. They dare not drop the shields to beam anyone on board, so Scotty must beam Spock to him to rescue him. The main computer is down, so Scotty must set the controls by hand. How can he figure out how far Spock is from Kirk so he can set the power levels on the transporter correctly?

Kirk and Spock have communicators so that they can talk to Scotty. How do they let him know where they are? Fortunately, they can both look outside windows and see the same large red building (draw this on the board). They are good at estimating distance and compasses on their communicators will let them know the direction. With this information, the red shirts must calculate the distance.

### Activity: Student Conversations

The main purpose here is to help students gain a geometric understanding of $|\Vec r - \Vec r'|$. In order to say where something is, you must first say where it is with respect to a known something else–an origin, in this case the red building. $\Vec r_K$ is Kirk's position and $\Vec r_S$ is Spock's position. We find it convenient to use mneumonic supscripts (K and S) on the vectors.

Various red shirts can then come to the board to draw pictures and calculate the distance between Kirk and Spock.

Emphasize that $\Vec r -\Vec r'$ is a vector and that you can find the magnitude of any vector by taking the square root of the dot product of the vector with itself.

Emphasize that $|\rr-\rrp|$ is a scalar!

Some students will want to avoid the complexities of using $|\rr - \rrp|$ by putting one point at the origin, thus effectively setting $\rrp$ to zero. This is a good time to point out that this is in fact an excellent strategy when there is just one point of interest, but fails if there are more.

### Activity: Wrap-up

It's important to emphasize that ideally, one just stretches a tape measure between two objects to find the distance between them. As an alternative, one calculates the distance using the Pythagorean theorem from the rectangular coordinates of the objects. But coordinate independent equations in physics write this distance as $|\Vec r - \Vec r'|$. Students should understand the relationship between vector subtraction, the magnitude of a vector, and the Pythagorean theorem.

### Extensions

It may interest some advanced students that while $|\Vec r - \Vec r'|$ is independent of coordinates, $\Vec r$ is not, quite. Or, at least, it is not independent of origin. The dilemma of how to locate objects in an origin free manner is a challenge in general relativity.

This activity is part of a sequence of activities addressing Power Series and their application to physics. The following activities are part of this sequence.

- Preceding activity:
- Recall the Electrostatic Potential due to a Point Charge: This small whiteboard question has students recall the basic expression for the electrostatic potential due to a point charge which is used to begin a classroom conversation regarding what is meant by $\frac{1}{r}$.

- Follow-up activities:
- Calculating Coefficients for a Power Series: This small group activity has students work out the expansion coefficients of a familiar function, $\sin(\theta)$, which gives them more experience working with power series.
- Approximating Functions with a Power Series: This computer visualization activity using Mathematica (or Maple) fits power series approximations of a given function to an actual function which allows students to see where approximations are valid.
- Electrostatic Potential Due to Two Point Charges: This small group activity has students apply what they know about power series approximations to find a general expression and asymptotic solution to the electrostatic potential due to two point charges.