## Activity: Dot Product Review

AIMS Maxwell AIMS 21 Static Fields Winter 2021
This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.
What students learn Basic algebraic and geometric properties of the dot product.
• Media
• group Right Angles on Spacetime Diagrams

group Small Group Activity

30 min.

##### Right Angles on Spacetime Diagrams
Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.
• assignment Building the PDM: Instructions

assignment Homework

##### Building the PDM: Instructions
PDM Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 In your kits for the Portable Partial Derivative Machine should be the following:
• A 1ft by 1ft board with 5 holes and measuring tapes (the measuring tapes will be on the top side)
• 2 S-hooks
• A spring with 3 strings attached
• 2 small cloth bags
• 4 large ball bearings
• 8 small ball bearings
• 2 vertical clamp pulleys
• A ziploc bag containing
• 5 screws
• 5 hex nuts
• 5 washers
• 5 wing nuts
• 2 horizontal pulleys
To assemble the Portable PDM, start by placing the PDM on a table surface with the measuring tapes perpendicular to the table's edge and the board edge with 3 holes closest to you.
1. one screw should be put through each hole so that the threads stick out through the top side of the board. Next use a hex nut to secure each screw in place. It is not critical that they be screwed on any more than you can comfortably manage by hand.
2. After securing all 5 screws in place with a hex nut, put a washer on each screw.
3. Slide a horizontal pulley onto screws 1 and 2 (as labeled above).
4. On all 5 screws, add a wing nut to secure the other pieces. Again, it does not need to be tightened all the way as long as it is secure enough that nothing will fall off.
5. Using the middle wingnut/washer/screw (Screw 4), clamp the shortest of the strings tied to the spring.
6. Loop the remaining 2 looped-ends of string around the horizontal pulleys and along the measuring tape.
7. Using the string as a guide, clamp the vertical pulleys into place on the edge of the board.
8. Through the looped-end of each string, place 1 S-hook.
9. Put the other end of each s-hook through the hole in the small cloth bag.
Here is a poor photo of the final result, which doesn't show the two vertical pulleys. If you would like, you could view a video of the building process.
• group Leibniz Notation

group Small Group Activity

5 min.

##### Leibniz Notation
AIMS Maxwell AIMS 21 Static Fields Winter 2021 This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics; unlike standard Leibniz notation, this notation explicitly specifies constant variables. Students are guided in linking the variables from a contextless Leibniz-notation partial derivative to their proper variable categories.
• assignment Vectors

assignment Homework

##### Vectors
vector geometry AIMS Maxwell AIMS 21

Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

Which pairs (if any) of these vectors

1. Are perpendicular?
2. Are parallel?
3. Have an angle less than $\pi/2$ between them?
4. Have an angle of more than $\pi/2$ between them?

• group Outer Product of a Vector on Itself

group Small Group Activity

30 min.

##### Outer Product of a Vector on Itself

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
• group Vector Surface and Volume Elements

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
AIMS Maxwell AIMS 21

Integration Sequence

Students use known algebraic expressions for vector line elements $d\vec{r}$ to determine all simple vector area $d\vec{A}$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a more sophisticated vector approach to find surface, and volume elements.

• group Flux through a Cone

group Small Group Activity

30 min.

##### Flux through a Cone
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Integration Sequence

Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ .
• group Using $pV$ and $TS$ Plots

group Small Group Activity

30 min.

##### Using $pV$ and $TS$ Plots
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

Students work out heat and work for rectangular paths on $pV$ and $TS$ plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.
• assignment Cross Triangle

assignment Homework

##### Cross Triangle
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Use the cross product to find the components of the unit vector $\mathbf{\boldsymbol{\hat n}}$ perpendicular to the plane shown in the figure below, i.e.  the plane joining the points $\{(1,0,0),(0,1,0),(0,0,1)\}$.

• assignment Ring Function

assignment Homework

##### Ring Function
Central Forces Spring 2021 Consider the normalized wavefunction $\Phi\left(\phi\right)$ for a quantum mechanical particle of mass $\mu$ constrained to move on a circle of radius $r_0$, given by: $$\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}$$ where $N$ is the normalization constant.
1. Find $N$.

2. Plot this wave function.
3. Plot the probability density.
4. Find the probability that if you measured $L_z$ you would get $3\hbar$.
5. What is the expectation value of $L_z$ in this state?

## Instructor's Guide

### Prerequisite Knowledge

This swbq will not work unless most students in the class have seen and used the dot product before. Use it at the beginning of term if the students need a review. If some students in your class have not seen the dot product before, you can refer them to the section The Dot Product in our online text.

### Introduction

Prompt: On your small whiteboard, write one thing about the dot product.

(The generic term “one thing” in the prompt is important. Do not use a more specific word that cues students to give an algebraic definition or a drawing, etc.)

### Student Conversations

strategy:smallwhiteboard:dotproductclip2007.mov|example video

Walk around the room as students are answering this question and quickly pick up an example of each different representation or statement. Quickly order them in the order you would like to talk about them and prop them on the chalkboard tray. Pick up each one (or more than one if you are comparing them) and give whatever review “lecture” you would normally give. The students are far more invested if they see that you are talking about their answers and some students will even vie with each other to get you to choose their answer. You can add in any extra representations that the students haven't mentioned as you go along.

The most important thing to emphasize is that the professional physicist knows and uses all of these representations/facts.

Answers you are likely to see:

### Extensions

You may also want to assign the homework problem Tetrahedron which requires students to use both the algebraic and geometric definitions of the dot product to solve the problem successfully.

Write one thing you know about the dot product.

Learning Outcomes