From 561bac75579391c14e47eaccfabdf9eda98855da Mon Sep 17 00:00:00 2001
From: Thomas Albers <thomas@thomaslabs.org>
Date: Wed, 27 Jul 2022 18:13:20 +0200
Subject: Initial commit

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+#+TITLE:Cross Product
+#+SETUPFILE: ../math_options.org
+#+LATEX_HEADER: \usepackage{bm}
+#+LATEX_HEADER: \usepackage{mathtools}
+
+* Disclaimer
+This site as of now just a technology demonstration and its claims
+should not be taken as true (even though I myself am pretty confident
+they are)
+
+* Definition
+We shall define the cross product in terms of two properties we are
+interested in:
+
+- Distributive: $\bm{a} \times \left(\bm{b} + \bm{c}\right) = \bm{a} \times \bm{b} + \bm{a} \times \bm{c}$
+- Orthogonal: $\bm{a} \times \bm{b} = \bm{c} \implies \bm{c} \cdot \bm{a} = \bm{0} \land \bm{c} \cdot \bm{b} = \bm{0}$
+
+It is worth mentioning that given a pair of vectors in $\mathbb{R}^3$
+there exist an infinite amount of vectors that satisfy these
+properties, so it is also necessary to introduce the following
+relations between the basis vectors to properly define the cross
+product.
+
+\begin{align*}
+\bm{e}_1 &= \bm{e}_2 \times \bm{e}_3\\
+\bm{e}_2 &= \bm{e}_3 \times \bm{e}_1\\
+\bm{e}_3 &= \bm{e}_1 \times \bm{e}_2\\
+\bm{e}_i \times \bm{e}_i &= \bm{0},\qquad\text{For}\quad i = 1,2,3
+\end{align*}
+
+We introduce the Levi-Civita symbol to condense our calculations.
+
+\begin{equation*}
+\epsilon_{ijk} \coloneqq \bm{e}_i \cdot \left(\bm{e}_j \times \bm{e}_k\right)
+\end{equation*}
+
+Based on this we may now derive a way to compute the cross product of
+two vectors
+
+\begin{align*}
+\left[\bm{a} \times \bm{b}\right]_i &= \left[\left(\sum_j a_j \bm{e}_j\right) \times \left(\sum_k b_k \bm{e_k}\right)\right]_i\\
+&= \left[\sum_{jk} a_j b_k \left(\bm{e}_j \times \bm{e}_k\right)\right]_i\\
+&= \sum_{jk} a_j b_k \bm{e}_i \cdot \left(\bm{e}_j \times \bm{e}_k\right)\\
+&= \sum_{jk} \epsilon_{ijk} a_j b_k\\
+\bm{a} \times \bm{b} &= \sum_{ijk} \epsilon_{ijk} a_i b_j \bm{e}_k
+\end{align*}
+
+* Properties
+\begin{align*}
+&\bm{a} \times \bm{b} = - \bm{b} \times \bm{a}\\
+&\bm{a} \cdot \left(\bm{b} \times \bm{c}\right)
+= \bm{b} \cdot \left(\bm{c} \times \bm{a}\right)
+= \bm{c} \cdot \left(\bm{a} \times \bm{b}\right)\\
+&\bm{a} \times \left(\bm{b} \times \bm{c}\right)
+= \left(\bm{a} \cdot \bm{c}\right)\bm{b} - \left(\bm{a} \cdot \bm{b}\right)\bm{c}\\
+&\left(\bm{a} \times \bm{b}\right) \times \left(\bm{c} \times \bm{d}\right)
+= \left(\bm{a}\cdot\bm{c}\right) \left(\bm{b}\cdot\bm{d}\right) - \left(\bm{a}\cdot\bm{d}\right) \left(\bm{b}\cdot\bm{c}\right)
+\end{align*}
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