Fundamental theorem

After a lecture introducing the cross product of vectors in mathbb{R}^3, student R countered me thus: “Dr. K, I’m a little worried about this cross product stuff.”

“How so?” I asked.

“Well, check this out,” R said.   He then proceeded to diagram the determinant expansion formula for the cross product:

mathbf{u} times mathbf{v} = left| begin{array}{ccc} mathbf{i} & mathbf{j} & mathbf{k}  u_1 & u_2 & u_3  v_1 & v_2 & v_3 end{array} right| = left| begin{array}{cc} u_2 & u_3  v_2 & v_3 end{array} right| mathbf{i} - left| begin{array}{cc} u_1 & u_3  v_1 & v_3 end{array} right| mathbf{j} + left| begin{array}{cc} u_1 & u_2  v_1 & v_2 end{array} right| mathbf{k}.

R continued: “When we work it out, we have to work out each of these little boxes of numbers–”

“2 x 2 determinants,” I offered helpfully.

“Right, each of these determinants by multiplying these two together and then subtracting these two multiplied together, like so.”   He indicated each with a swooshing motion:

“Is it just me,” R said ,”or does that look just like a Jesus fish?”

“Er,” I said.   The terms coincidence and conspiracy nut briefly flashed in my mind.

“And you get the cross product,” R continued, ” by computing three of these little things.   And you said that any two perpendicular vectors can be crossed to get a third vector to form a ‘local frame of reference.’   So the cross product is one of three things that together form your frame of reference.   Doesn’t that sound just like the Holy Trinity?”

“Huh,” I replied.   I hadn’t really ever thought of that.

“And even the name of thing — the cross product?   The symbol of Christianity?”

I was speechless.

“So,” R concluded, “isn’t all this vector stuff really just a plug for Christianity?   And if so, shouldn’t I be excused from being forced to take calculus on First Amendment grounds?”


The next day in class, I pulled R aside.

“You raise an interesting point yesterday,” I admitted.   “However, your fears are entirely unfounded.   If you go on a little further in mathematics and take a course in Advanced Calculus, you’ll discover first that all the basic operations of calculus are justified using   so-called epsilondelta proofs, which are more or less arguments built up by comparing things to almost nothing.   Go a little further, and you’ll even discover that all of the numbers we use in calculus are built from the empty set, which is quite literally nothing. Hence, calculus is really teaching you that one of the most effective ways to understand the world around you is to accept that it’s inherently based on nothingness.”

R considered this a moment.

“So,” I concluded, “I’m actually apparently plugging Buddhism instead.”

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