Vector Formalism in Introductory Physics VI: A Unified Solution for Simple Dot Product and Cross Product Equations

TL;DR: Simple vector dot products and cross products may be “undone” using formal methods consistent with Gibbsian vector algebra. Writing the cross product and dot product of an unknown vector relative to a given vector in a canonical form allows a well known vector identity to be used to isolate the unknown vector. Special cases […]

Read More Vector Formalism in Introductory Physics VI: A Unified Solution for Simple Dot Product and Cross Product Equations

Vector Formalism in Introductory Physics V: Two Equations, One Solution

TL;DR: Solving seemingly trivial dot product and cross product equations leads to an astonishing result, namely that they have the same solution, which can be derived both geometrically and algebraically. Establishing this common solution is an important step in motivating formal Gibbsian vector algebra. In the previous two posts, I demonstrated that the simple dot […]

Read More Vector Formalism in Introductory Physics V: Two Equations, One Solution

Vector Formalism in Introductory Physics IV: Unwrapping Cross Products Geometrically

TL;DR: Vector cross products are not like products of real numbers, for which there is an inverse operation to “undo” multiplication. I don’t think we should introduce cross products as a form of “multiplication” in introductory physics courses because it may reinforce the urge to “divide by a vector.” A better approach may be to […]

Read More Vector Formalism in Introductory Physics IV: Unwrapping Cross Products Geometrically

Vector Formalism in Introductory Physics III: Unwrapping Dot Products Geometrically

TL;DR: Vector dot products are not like products of real numbers, for which there is an inverse operation to “undo” multiplication. I don’t think we should introduce dot products as a form of “multiplication” in introductory physics courses because it may reinforce the urge to “divide by a vector.” A better approach may be to […]

Read More Vector Formalism in Introductory Physics III: Unwrapping Dot Products Geometrically

Angular Quantities II

In this post, I will address the first question on the list in the previous post. What exactly does it mean for something to be a vector? In almost every introductory physics course, vectors are introduced as “quantities having magnitude and direction” and are eventually equated to graphical arrows. A vector is neither of these, […]

Read More Angular Quantities II

Matter & Interactions II, Week 7

This week, I was away at the winter AAPT meeting in Atlanta. Students began working on the experiments from chapter 17, which serve to introduce magnetic fields. I want to emphasize some really cool things about the mathematical expression for a particle’s magnetic field: This is really a single particle form of the Biot-Savart law. […]

Read More Matter & Interactions II, Week 7

Matter & Interactions II, Week 6

I’m writing this a whole week late due, in part, to having been away at an AAPT meeting and having to plan and execute a large regional meeting of amateur astronomers. This week was all about the concept of electric potential and how it relates to electric field. I love telling students that this topic […]

Read More Matter & Interactions II, Week 6

Matter & Interactions I, Week 11

As (almost) usual, I’m writing this on the Monday after the week in question. This week we hit chapter 5, which is packed full of interesting physics and mathematics! We encounter the infamous time derivative of a unit vector (aka a direction), which I have found quite mysterious because of the rather hand waving ways […]

Read More Matter & Interactions I, Week 11

Matter & Interactions I, Week 7

I will edit this post at a later date to include diagrams. This was a pretty good week. Students are beginning to ask deeper, more insightful questions and that tells me they’re engaging in the material more than before. We formally began chapter 2 on the momentum principle, but I did something quite different from […]

Read More Matter & Interactions I, Week 7

Matter & Interactions I, Week 5

This week, we transitioned to chapter 1 of the Matter & Interactions textbook (fourth edition). I have WebAssign problem sets for each chapter available for formative assessment and practice while working their way through the reading. I encouraged them to use the book the way it was intended to be used, specifically by stopping and […]

Read More Matter & Interactions I, Week 5