Lagrange's formula (vector triple product) for complex vectors

Question

I am currently reading into light scattering theory and trying to derive the formula for the extinction cross section. At some point I have the expression $a\times(b\times c)$, where $a,b,c\in\mathbb{C}^3$ (three dimensional complex vectors). A quick google search reveals the wikipedia article about the triple product, but testing this out with Matlab gives two different results, i.e., $a\times(b\times c) \neq (a\cdot c)b - (a\cdot b)c$. Fiddling around with it, I was able to produce the formula $a^\ast\times(b\times c) = (a\cdot c)b - (a\cdot b)c$, where $a^\ast$ is the complex conjugate of $a$. My question is if this formula is correct and if there are some rules for the complex conjugate of a cross product. I assumed and tested that $(b\times c)^\ast = b^\ast \times c^\ast$.

Answer 1

The confusion comes from the fact that there are multiple ways to define "$a\cdot b$"

1. define $a\cdot b=\sum_i a_i b_i$
2. define $a\cdot b=\sum_i a_i^* b_i$
3. define $a\cdot b=\sum_i a_i b_i^*$

then

• for real vectors they are all the same
• the formula you quote is only valid for (1). Because the cross-product does not include any complex conjugation.
• (2) is the most common in mathematics, because that way it defines a "scalar product" (in the sense of a Hilbert space)
• (3) is an alternative definition of scalar product / hilbert space. I think this is less common. But it is just a convention.
• some computer algebra systems implement option (1), you have to type the complex conjugation yourself if you want it.