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 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. […]