## Activity: Dot Product Review

Static Fields 2022 (6 years)
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

Write one thing you know about the dot product.

## 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 you know 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.

• group Right Angles on Spacetime Diagrams

group Small Group Activity

30 min.

##### Right Angles on Spacetime Diagrams
Theoretical Mechanics (4 years)

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.
• group Vector Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

• keyboard Sinusoidal basis set

keyboard Computational Activity

120 min.

##### Sinusoidal basis set
Computational Physics Lab II 2022

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
• assignment Building the PDM: Instructions

assignment Homework

##### Building the PDM: Instructions
PDM Energy and Entropy 2021 (2 years) 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 Projectile with Linear Drag

group Small Group Activity

120 min.

##### Projectile with Linear Drag
Theoretical Mechanics (4 years)

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
• assignment Free energy of a harmonic oscillator

assignment Homework

##### Free energy of a harmonic oscillator
Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020

A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with $\varepsilon_n = n\hbar\omega$, where $n$ is an integer $\ge 0$, and $\omega$ is the classical frequency of the oscillator. We have chosen the zero of energy at the state $n=0$ which we can get away with here, but is not actually the zero of energy! To find the true energy we would have to add a $\frac12\hbar\omega$ for each oscillator.

1. Show that for a harmonic oscillator the free energy is $$F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right)$$ Note that at high temperatures such that $k_BT\gg\hbar\omega$ we may expand the argument of the logarithm to obtain $F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)$.

2. From the free energy above, show that the entropy is $$\frac{S}{k_B} = \frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1} - \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right)$$

This entropy is shown in the nearby figure, as well as the heat capacity.

• assignment Free energy of a two state system

assignment Homework

##### Free energy of a two state system
Helmholtz free energy entropy statistical mechanics Thermal and Statistical Physics 2020
1. Find an expression for the free energy as a function of $T$ of a system with two states, one at energy 0 and one at energy $\varepsilon$.

2. From the free energy, find expressions for the internal energy $U$ and entropy $S$ of the system.

3. Plot the entropy versus $T$. Explain its asymptotic behavior as the temperature becomes high.

4. Plot the $S(T)$ versus $U(T)$. Explain the maximum value of the energy $U$.

• group Generalized Leibniz Notation

group Small Group Activity

10 min.

##### Generalized Leibniz Notation
Static Fields 2022 (4 years) 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.
• group Earthquake waves

group Small Group Activity

30 min.

##### Earthquake waves
Contemporary Challenges 2022 (4 years)

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.
• assignment Vectors

assignment Homework

##### Vectors
vector geometry Static Fields 2022 (3 years)

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?

Learning Outcomes