Conceptual Understanding in Introductory Physics VII: Vector Division

This post begins a series of questions centered on vector analysis. I have always felt that vectors get insufficient treatment in introductory physics courses. They’re presented too quickly, with too little attention paid to consistent notation and practically no attention paid to their analytical and geometrical properties. I try very hard to correct these oversights in my course.

Students learn early on in mathematics that division by zero is undefined, yet very few students can explain what that actually means. Similarly, students are told (note the choice of words) in either mathematics or physics that division by a vector is also undefined. I’ve yet to find a student who can correctly articulate why this is. So, naturally this is the first question in this series.

Explain why in traditional Gibbsian vector algebra division by a vector is not defined. You may include diagrams if they help the clarity of your explanation, but you must articulate a non-diagrammatic explanation.

When introducing vectors, I strongly recommend that you also tell students the history behind vector notation and how the current framework of vector analysis came to be despite its inferiority to other frameworks like geometric algebra. The story has all the elements of contemporary academia: politics, ego, popularity contests, and cutthroat competition, and intrigue. Students should also get or be given a PDF copy of the first contemporary textbook on vector analysis by Wilson, based on Gibbs’ work.

This series of questions should generate lots of discussion. Go!