Multiple cross products Guide, Meaning , Facts, Information and Description

In mathematics, there are tricks for multiple cross products. The cross product operation is not associative. On the other hand exterior algebra is associative. Therefore this is an aspect of multilinear algebra that goes beyond the basics of the wedge product.

The product (A X B) X C doesn't equal A X (B X C), in general. There are two different basic operations, crossing on the left and crossing on the right. This article develops the operation for the right. This is sufficient since the cross product is anticommutative: left and right can be switched with a change of sign, at most.

Normally something like

A X ((B X C) X D)

might be calculated by utilising the determinant pattern three times. Doing the determinants over and over is the complicated way: they can be worked out directly in the following way.

A = (a₁, a₂, a₃) = a^<1>.

The last term means each component has an a, the first component is numbered 1 and the numbers go up 1 for each component.

So

A X B = (a2.b3 - a3.b2 , a3.b1 - a1.b3 , a1.b2 - a2.b1) = ab^<23-32>

That is, every term in every component has an a and b in it so they go out the front in order, then you use the pointed bracket to show how the index numbers are distributed, and an up arrow to show how to get from the first component to the second to the third. Addition cycles around the index numbers:

1+1 = 2 , 2+1 = 3, 3+1 = 1.

That is, 1-2-3 is considered a cyclic order.

Repeated crossing on the right produces a sequence of vectors

A , A X B, (A X B) X C , ((A X B)X C)X D, ...

where

A = a<1>¤

A X B = ab<23 - 32>¤

(A X B) X C = abc< 313 - 133 - 122

+212>¤

( A X B) X C) X D = abcd< 1213 -

2113 - 2333 + 3233 - 2322 + 3222 + 3112 - 1312>¤

Aside from being a way of avoiding the need to write the other components, the pointy bracket's main purpose is to facilitate a straightforward operation of converting a vector into a vector crossed on the right, in other words we are going to use it to find an operation R such that:

R(A) = A X B, R(A X B) = (A X B) X C , etc.,

and the R( ) that turns out to work is the one that adds an additional letter on the right just before the pointy bracket, lowers each index number of each index term by 1 and adds an additional term of -2 to each term on the right, then raises each index number of each index term by 1 and puts a 3 at the end of each term. Denote it by

R( ) = 3/-2]( )

so the lower side of the slash is the number you put at the end of the lowered index terms and the upper is the one for the raised terms accordingly.

An example to check this all.

R(A X B )

= 3/-2](A X B )

=3/-2](ab<23 - 32>¤)

=abc 3/-2](<23 - 32>¤)

=abc<31 3- 13 3+ -12 2- - 21 2>¤

=313 - 133 - 122 + 212 >¤

= (A X B) X C

This is an Article on Multiple cross products. Page Contains Information, Facts Details or Explanation Guide About Multiple cross products

	Web www.E-paranoids.com