This post’s question reinforces the previous question in this series, which asks the reader to articulate why division by a vector is undefined in Gibbsian vector analysis. Here’s the question:
Solve the following vector equation for a:
Don’t be deceived. It’s not as straightfoward as it looks. I would really love hearing from colleagues who give this problem to their students. I predict most (students) will try to divide both sides by a vector to get the answer.
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!
In this post, I present a question from special relativity that addresses how we name reference frames. Students tend to blindly memorize names, terms, labels, and other minutiae that have little or nothing to do with the underlying physics.
In one problem, a clock is stated to be in the S frame and has a tick-tock of a certain duration. In another problem, the same clock is stated to be in the S’ frame and has a tick-tock of a shorter duration. Is there any danger in generalizing a clock’s measured behavior as always being associated with the S or S’ frame? In other words, articulate the repercussions, if there are any, of saying, “A clock in the S’ frame will always have a shorter tock-tock than a clock in the S frame.”
When I have given my own students this question (and I don’t use this question every semester) they have nearly always articulated a good response. How would your own introductory students fare?
In the next post, I will begin a series of conceptual questions themed around vector analysis.