Conceptual Understanding in Introductory Physics XXXVI: Geometry and Work

This question is particularly revealing in that it assesses whether or not students understand the coordinate-free nature of work as a dot product. Be prepared to hear such nonsense locutions as “negative force” or “negative displacement” but don’t be shocked when you hear them. I think it’s a product of the relatively poor treatment vectors […]

Conceptual Understanding in Introductory Physics XXIX: Two Expressions for the Dot Product

Students sometimes see vector dot products in their calculus classes before they see them in their physics classes. Dot products are often presented with two seemingly unrelated definitions, one of which is geometric and coordinate free and the other is in terms of components in a particular basis. Yet, the two give exactly the same […]

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 […]

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 […]

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 […]

Vector Formalism in Introductory Physics II: Six Coordinate-Free Derivations of the BAC-CAB Identity

TL;DR: The BAC-CAB vector identity is probably the most important vector identity, and has potentially important applications in introductory physics. I present six coordinate-free derivations of this identity. By “coordinate-free” I mean a derivation that doesn’t rely on any particular coordinate system, and one that relies on the inherent geometric relationships among the vectors involved. […]

Vector Formalism in Introductory Physics I: Taking the Magnitude of Both Sides

TL;DR: I don’t like the way vectors are presented in calculus-based and algebra-based introductory physics. I think a more formal approach is warranted. This post addresses the problem of taking the magnitude of both sides of simple vector equations. If you want the details, read on. This is the first post in a new series […]

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 […]

Matter & Interactions I, Week 13

As usual, I’m posting this the Monday after the week named in the title. This week was all about chapter 6: energy and the energy principle. This is where Matter & Interactions really shines among introductory textbooks. I remember as a student being so confused by sign conventions that I honestly never knew when to […]

Did Feynman Invent Feynman Notation?

In section section 27-3 of The Feynman Lectures on Physics, Feynman describes a notation for manipulating vector expressions in a way that endows nabla with the property of following a rule similar to the product rule with which our introductory calculus students are familiar. It allows a vector expression with more than one variable to be […]