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 […]

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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, […]

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. […]

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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 […]

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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 […]

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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 […]

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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 […]

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Conceptual Understanding in Introductory Physics XXV

This question emphasizes geometry and should be done without use of a coordinate system. It should also be done using only symbolic manipulation of vectors. Here it is. Consider a particle moving with a constant, non-relativistic velocity. Starting with a general expression for kinetic energy in terms of either velocity or momentum, prove that the […]

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Help Me! Question About Tensors and Projections

I need some help. I am working hard to find ways to bring more geometry into introductory calculus-based physics (and conceptual physics as well). By geometry, I mean specifically the geometry associated with vectors and tensors, and the information encoded therein. Yes, I said tensors. I have been heavily influenced by these notes by Kip […]