Conceptual Understanding in Introductory Physics XXX: Two Expressions for the Cross Product

I just realized this is the thirtieth post in this series! I don’t know if anyone has found this series helpful, but I think these questions collectively might make a pool of original exam questions. That’s mainly how I see them anyway.

This post is almost a word for word duplicate of the last post in this series, with a few obvious differences.

Students sometimes see vector cross products in their calculus classes before they see them in their physics classes. Magnitudes of cross products are often presented with two seemingly unrelated definitions, one of which is geometric and coordinate free and the other is in terms of components in a particular basis. Yet, the two give exactly the same number, which is rather counterintuitive.

Given two vectors \mathbf{a} and \mathbf{b}, the magnitude of their cross product is given by \left\lVert\mathbf{a}\times\mathbf{b}\right\rVert = \left\lVert\mathbf{a}\right\rVert \left\lVert\mathbf{b}\right\rVert \sin\theta, which in words is just the product of the vectors’ magnitudes and the sine of the angle between them. But students also learn that the squared magnitude of the cross product can be calculated in terms of the scalar components as \left\lVert\mathbf{a}\times\mathbf{b}\right\rVert^2 = (a_2 b_3 - a_3 b_2)^2 + (a_3 b_1 - a_1 b_3)^2 + (a_1 b_2 - a_2 b_1)^2, which in words is just the sum of the squares of antisymmetric pairwise products of corresponding scalar components of the two vectors (that was a mouthful and I hope I said it correctly!).

Here’s the question: Show that these two expressions for \left\lVert\mathbf{a}\times\mathbf{b}\right\rVert^2 are equal; show that \left\lVert\mathbf{a}\right\rVert^2 \left\lVert\mathbf{b}\right\rVert^2 \sin^2\theta = (a_2 b_3 - a_3 b_2)^2 + (a_3 b_1 - a_1 b_3)^2 + (a_1 b_2 - a_2 b_1)^2. (HINT: Remember that the cross product represents a parallelogram, the area of which is the cross product’s magnitude. Make a drawing and use the result of the last post.)

I think this question is extremely important because it connects two concepts that at first glance look totally unrelated. No one ever showed me this as a student, and indeed I doubt very seriously that most introductory physics students have seen it.


Feedback is welcomed as always!

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