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 […]

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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 […]

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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 […]

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Proving the Absence of Length Contraction Perpendicular to Velocity

When teaching conceptual physics, students almost always ask why length contraction only occurs parallel to velocity and not perpendicular to it. That’s a meaty conceptual question and one that always leaves me looking for a convincing, non mathematical explanation. Earlier this semester, I finally found what I think is the best one I’ve ever heard […]

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Musical Thoughts On Teaching Physics

TL;DR There are many lessons physics teachers can learn from music teachers about teaching one’s discipline. Many, and perhaps most (all?), concepts in music have analogs in physics and mathematics. I have a background in music, spefically percussion. Marching band was my life in high school and that carried over into my college years. I […]

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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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´╗┐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. […]

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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 […]

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