Conceptual Understanding in Introductory Physics XXXI: Practice With Vector Notation

I’m going to present a series of questions I used this past semester for oral interviews┬áin my calculus-based mechanics (Matter & Interactions) course. These questions are intended to go beyond merely working problems. I think you will find that students who appear to do well at rote problem solving will have difficulty with many of these questions. I also want questions for which simple answers aren’t likely to be found online, particularly on certain popular websites (popular with students who cheat, that is). These questions are structured so that you may use many variations of each one. There’s no rule that each student must get the exact same version of every question. In practice, I asked students to do this using the interactive whiteboard in Zoom. I chose random items for each student to hopefully discourage cheating.

Consider an arbitrary vector, say \mathbf{a}. Write the mathematical symbol for the quantity defined by each phrase below.

  • the magnitude of \mathbf{a}
  • the direction of \mathbf{a}
  • the time derivative of \mathbf{a}
  • the time derivative of the magnitude of \mathbf{a}
  • the time derivative of the direction of \mathbf{a}
  • the magnitude of the direction of \mathbf{a}
  • the magnitude of the time derivative of the direction of \mathbf{a}
  • the opposite of the direction of \mathbf{a}
  • the z-component of \mathbf{a}
  • the Cartesian coordinate representation of \mathbf{a} in symbols, not numbers

This question should be done at the end of chapter 1 (or the chapter dealing with vectors). Later in the course, you can add things like “the dot product of \mathbf{a} with…” or “the cross product of \mathbf{a} with…” to reinforce those notations. Coordinate-free vector notation is extremely important and, unfortunately, not treated consistently in introductory courses, even M&I. Students must be as proficient as they should be in other mathematical notation from their other courses.

A variation of this question might be to show the student a symbol and ask them to say in words what the symbol means and/or to write out what it means.

As always, feedback is welcome.

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