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Hometriple cross product

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# triple cross product

The cross product of a vector with a cross product is called the triple cross product.

The expansion formula of the triple cross product or Lagrange’s formula is

$\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot% \vec{b})\vec{c}$ |

(“exterior dot far times near minus exterior dot near times far” — this works also when “exterior” is the last factor).

The formula shows that this vector is in the plane spanned by the vectors $\vec{b}$ and $\vec{c}$ (when these are not parallel).

Note that the use of parentheses in the triple cross products is necessary, since the cross product operation is not associative, i.e., generally we have

$(\vec{a}\times\vec{b})\times\vec{c}\neq\vec{a}\times(\vec{b}\times\vec{c})$ |

(for example: $(\vec{i}\times\vec{i})\times\vec{j}=\vec{0}$ but $\vec{i}\times(\vec{i}\times\vec{j})=-\vec{j}$ when $(\vec{i},\,\vec{j},\,\vec{k})$ is a right-handed orthonormal basis of $\mathbb{R}^{3}$). So the system $(\mathbb{R}^{3},\,+,\,\times)$ is not a ring.

A direct consequence of the expansion formula is the Jacobi identity

$\vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{c}\times\vec{a})+\vec{c% }\times(\vec{a}\times\vec{b})=\vec{0},$ |

which is one of the properties making $(\mathbb{R}^{3},\,+,\,\times)$ a Lie algebra.

It follows from the expansion formula also that

$(\vec{a}\times\vec{b})\times(\vec{c}\times\vec{d})=(\vec{a}\vec{b}\vec{d})\vec% {c}-(\vec{a}\vec{b}\vec{c})\vec{d}$ |

where $(\vec{u}\vec{v}\vec{w})$ means the triple scalar product of $\vec{u}$, $\vec{v}$ and $\vec{w}$.

## Mathematics Subject Classification

15A72*no label found*

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## Comments

## triple cross product

The expansion formula agrees with a simple nemotechnic rule:

"exterior dot far times near minus exterior dot near times far".

That rule works if $\vec{a}$ is in left or right side.

## Re: triple cross product

Thank you, it is a good mnemonic -- I think I add it to the object, if you give permission!

Jussi

## Re: triple cross product

Dear Mr. pahio,

Of course yes!! For me is an honor.

Best regards,

Pedro

## triple vector cross product

Hi, It would be nice to have some geometric proof of this for offering some insight.

## "geometric proof"

See http://it.wikipedia.org/wiki/Prodotto_misto (Semplificazioni) and http://es.wikipedia.org/wiki/Doble_producto_vectorial (Demostración).