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Triple Vector Product Order

As it wasn't immediately obvious to me that

\mathbf{x \times (y \times z) \neq (x \times y) \times z}

(perhaps it ought to have been, but hey) I decided to thrash it out in ink. I found the aforementioned is in fact true, but I couldn't figure out a nice expression for the difference, which has left me rather irritated.

Denoting

\mathbf{x} = (a,b,c),\ \mathbf{y}=(d,e,f),\ \mathbf{z}=(g,h,k)

and defining

\displaystyle \mathbf{(x \times y) \times z - x \times (y \times z)} = \Delta

I found that:

\Delta_i = -a(eh+fk) + g(be+cf)

\Delta_j = -b(dg+fk) + h(ad+cf)

\Delta_k = -c(dg+eh) + k(ad+be)

So my question is, is there a nice vector identity for

\Delta

in terms of the individual vectors?

Reply 1

ghostwalker

Original post by Astronomical

So my question is, is there a nice vector identity for

\Delta

in terms of the individual vectors?

Cribbed from wiki - no proof, but should be easy enough.

Untitled.jpg3.2KB

Reply 2

Astronomical

Original post by ghostwalker

Cribbed from wiki - no proof, but should be easy enough.

Not sure why it didn't occur to me to check good ol' wikipedia. Thanks!

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Triple Vector Product Order

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