blob: ceab6914d0278ee1a1e3b7b11471c7a771cbf967 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
|
#+title:Cross Product
#+setupfile: ../../math_options.org
* 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*}
|
