If we choose to define intelligence as the ability to creatively make associations, then questions are the father of intelligence. There is no insight without inquiry. Without the uniquely human faculties of imagination and curiosity, it is hard to imagine that humanity would have progressed far beyond the cave. Among other things, there would be no internet, no internal combustion engine, and no vestige of us on the moon. Even our omniscient and omnipotent AI overlords require methodical prompting in the form of human questions to enhance their performance.

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Perhaps this implies that we live in a world that encourages exploration and investigation, and it seems to, albeit on a superficial level. We have all heard at one point or another that there is no such thing as a stupid question. Another good aphorism says we should trade brilliance for curiosity. Maybe we remember that seventh grade science teacher who inspired our innate human curiosity or recall a parent telling us to always ask questions. Despite all our technological progress and emphasis on inquiry, these sentiments go right out of the window when we are learning mathematics.

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Humanity tends to fear what it doesn’t know while clinging to what it postulates arbitrarily. Sometimes, these are even the same thing. Take the Pythagorean theorem, for example. Anyone learning middle school math can tell us that $a^2 + b^2 = c^2$. But if we ask around why this is true, we are met with wide eyes and mouths agape. In an age where we have the frontiers of knowledge at our fingertips, there are still some questions that we just don’t ask.

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This isn’t entirely our fault though. I blame the public school system. Back in the 1960s, the USA was in a space race with the Soviets. This led to the wide adoption of a math curriculum aimed at cranking out the next generation of engineers to the detriment of studying the foundations, like set theory—a branch of math essential for learning how to prove mathematical statements. This shift partially robbed us of an intellectual inheritance that took thousands of years to accrue, an inheritance that educators had been working hard to give us only a decade before.

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Such a change is detrimental for more reasons than just passing ignorance. I have heard from more than one mathematician, and an engineer, that proofs are what make mathematics so satisfying and beautiful to them. As something of an aficionado myself, I remember falling in love with proof-based math at the age of 25, from prison! I sat at the same dayroom table reading Geometry: The Easy Way for a whole month, doing every problem. In a world where math scores are declining, I ask: how many mathematicians, how many physicists, how many engineers are out there not fulfilling their roles because of our curricular emphasis on facts instead of reasoning?

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I see a plausible solution. We should all start asking more questions. Let’s make our curiosity contagious and annoy beyond belief those who don’t catch the bug. Remember when you were young and asked your uncle “why” a million times before he yelled at you? Yes, that curious!

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Sometimes, relentlessly seeking out an answer can lead to amazing conclusions. A few months ago, I wanted to find an explanation for Lagrange’s cross product formula that wasn’t computational, but that still fit into the framework of basic linear algebra. Without access to the internet or any resources that could confirm this, this short proof was an endeavor! For those interested, that proof follows for $v(u \cdot w) - w(u \cdot v) = u \times (v \times w)$ such that $u,v,w \in \mathbb R^3$.

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Proof: The cross product outputs a vector that is orthogonal to the plane spanned by the two input vectors. This means $(v \times w) = h$ such that $h^t(cv + dw) = 0$. The left side of our equation is simplified to $u \times h$.

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Taking the cross product again, we get another vector that is orthogonal to ${\rm Span}(u,h)$, call it $q$. Since this vector q is orthogonal to the vector $h$ —the orthogonal compliment of ${\rm Span}(v,w)$— it must be in the plane ${\rm Span}(v,w)$.

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That means there is a set of linear combinations such that $cv + dw = kq , (c,d,k \in \mathbb R)$. This linear combination can be found by noting the fact that

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$$u^{T}h = u^{T}(cv + dw) = u^{T}cv + u^{T}dw = 0.$$

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Shifting some stuff around we see that

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$$-\frac{d}{c} = \frac{u^{T}v}{u^{T}w}$$

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Any two scalars with this negative ratio will suffice to fall on the line $kq$. That means that these scalars will work themselves as $c$ and $d$.

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We now have

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$$v(u \cdot w) - w(u \cdot v) = u \times (v \times w) = q$$

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q.e.d.

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