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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*}