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 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 […]

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 […]

Angular Quantities I

This is the first in a series of posts in which I want to share some hopefully interesting things about mathematical descriptions of rotational motion. This series was inspired by a talk given at the 2015 winter AAPT meeting in San Diego. The author claimed to have found a way to represent angular displacement as […]

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 […]

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 […]

Conceptual Understanding in Introductory Physics XXV: A Free Particle?

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 […]