# Cross product associative triples

• nikcs123
In summary: I think I know what you are getting at. I understand what you are saying. But I don't think you are right.In summary, the cross product is not associative in general, but certain special triples of vectors can satisfy the identity. These triples are known as "associative" and can be characterized by the condition that the vectors are either parallel or equal. However, this conclusion may have been reached prematurely and it may be necessary to consider other factors, such as the scalar product, to fully characterize all nonzero associative triples.
nikcs123
cross product "associative triples"

## Homework Statement

We know that the cross product is not associative, i.e., the identity
(1) $$(\vec{a}\times\vec{b})\times\vec{c}$$ = $$\vec{a}\times(\vec{b}\times\vec{c})$$ is not true in general. However, certain special triples $$\vec{a}$$;$$\vec{b}$$;$$\vec{c}$$
of vectors do satisfy (1). For example, if one of the vectors is the zero vector, then (1)
holds trivially, but there are also less obvious examples. Call a triple $$\vec{a}$$;$$\vec{b}$$;$$\vec{c}$$ for which (1)
holds "associative". Characterize all nonzero associative triples by some simple geometric
condition. (An example of a possible condition (though not the correct one) would be that
the three vectors are pairwise perpendicular.)

## The Attempt at a Solution

I attempted to use the properties/identities of cross products to deduce a relationship between the components of each vector and try to piece it together that way, ended up with a huge mess and no progress... Just need a nudge in the right direction on this one.

Last edited:

Did you try to use the triple product expansion formulas:

$$a\times(b\times c)=b(a,c)-c(a,b)$$

$$(a\times b)\times c=-c\times (a\times b)=c\times (b\times a)=...$$

$$(\vec{a}\times\vec{b})\times\vec{c}=\vec{a}\times(\vec{b}\times\vec{c})$$

Working with the left side,

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

So then

$$\vec{c}\times(\vec{b}\times\vec{a})=\vec{a}\times(\vec{b}\times\vec{c})$$

For the above to be true, $$\vec{a}=\vec{c}$$. Furthermore, the above also holds true when $$\vec{a}$$ and $$\vec{c}$$ are parallel.

That is correct, right?

nikcs123 said:
For the above to be true, $$\vec{a}=\vec{c}$$. Furthermore, the above also holds true when $$\vec{a}$$ and $$\vec{c}$$ are parallel.

That is correct, right?

Why don't you bring in the equality involving scalar products and justify your conclusion? Perhaps you have missed something?

## 1. What is a cross product associative triple?

A cross product associative triple is a mathematical concept used to describe the association of three elements in a set or vector space. It is commonly used in vector algebra and is closely related to the concept of the cross product of two vectors.

## 2. How is a cross product associative triple calculated?

A cross product associative triple is calculated by taking the cross product of two vectors, then taking the cross product of the resulting vector with a third vector. This is commonly written as (a x b) x c, where a, b, and c are the three vectors in the triple.

## 3. What is the significance of cross product associative triples?

Cross product associative triples are important in mathematics and physics because they allow us to describe the relationship between three vectors in a vector space. They are also used in calculations involving torque, angular momentum, and other physical quantities.

## 4. Can a cross product associative triple be used in higher dimensions?

Yes, a cross product associative triple can be used in any number of dimensions, although it is most commonly used in three dimensions. In higher dimensions, the calculation becomes more complicated, but the concept of associativity still holds.

## 5. Are there any real-world applications of cross product associative triples?

Yes, cross product associative triples have numerous real-world applications. They are used in physics to calculate the torque on a rotating object, in engineering to design machines and structures, and in computer graphics to calculate lighting and shading effects.

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