Why is the Vector Product Operation Considered Multiplication?

[Battlemage!]

The operation is called "the cross product." I am wondering if there are any specific characteristics that has led mathematicians to consider it a type of multiplication. After taking abstract algebra, the only conclusion I can come away with is that it more closely resembles traditional multiplication than traditional addition.

Are there any other compelling reasons?


Thank you!

 

[Fredrik]

Mathematicians don't like to use the + notation or call the operation "addition" when it's not commutative. For the cross product, it's not true in general that $x\times y=y\times x$. Actually, the cross product isn't even associative.

 

[jedishrfu]

Also we associate multiplication with finding areas and volumes and in the case of the cross product we associate it with the area of the parallelogram defined by vectors A and B.

 

[Battlemage!]

Ah, these are very solid answers. Thank you. Does anyone else have something they want to add?

 

[I like Serena]

Yep.
The length of the cross product is the product $a b \sin \theta$.
The dot product is $a b \cos \theta$.
Those really look like products (as opposed to additions).

 

[Number Nine]

Ah, these are very solid answers. Thank you. Does anyone else have something they want to add?

Part of it has to do with with the definition of vector space. A vector space, formally, includes two sets of objects; the first (the scalars) is called a field (if you don't know what that means, just think of the real numbers), and the second (the vectors) is what's called an abelian group under addition, which means that it already has an addition operation that behaves very nicely. The two sets together make up a vector space over the field (so taking the set of real numbers and the set of all n-tuples of real numbers with component-wise addition gives you an n-dimensional vector space over the reals). Since the vectors already have addition, it's fairly natural (but not really necessary) that you would relate new operations to multiplication in some way (of course, it helps that the definitions of both the dot and cross products actually involve multiplication).

One thing that you'll appreciate more when you take a good course in abstract algebra (highly recommended) is that there's nothing special about addition and multiplication, other than that they behave nicely with real numbers. You can define all sorts of new operations and call them whatever you want; all that matters is how they behave.

 

[SteveL27]

Ah, these are very solid answers. Thank you. Does anyone else have something they want to add?

Or multiply? :-)

 

[Battlemage!]

I suppose it was kind of an airhead question, after all the only binary operations available are addition and multiplication. Still, if I'm not mistaken, there are other things with vectors that we do that require the input of two vectors and yield one result that are not called multiplication (such as vector projection). But I think together these explanations you have all given are more than satisfying: (1) the choices were either addition or multiplication, (2) multiplication is often associated with area and volume, and the cross product is associated with the area of a parallelogram, (3) calculating their magnitudes is nothing but real number multiplication.

Altogether it does make sense. Thanks.

Or multiply? :-)

Ahahahah! Nicely done.

 

[I like Serena]

Hmm, you can define any operation.
Addition and multiplication are only the most common ones.
Often an abstract operation is denoted as $\circ$ or * to distinguish it from addition (+) or multiplication ($\times$ or $\cdot$).

In particular the dot product and cross product use 2 different representations of the multiplication operator.
Beyond that the "outer product" is defined which is often denoted as $\wedge$, but to be fair this is only a generalization of the cross product.

 

[Number Nine]

I suppose it was kind of an airhead question, after all the only binary operations available are addition and multiplication.

Not true. You can define any number of operations on any set you want.

 

[chiro]

The operation is called "the cross product." I am wondering if there are any specific characteristics that has led mathematicians to consider it a type of multiplication. After taking abstract algebra, the only conclusion I can come away with is that it more closely resembles traditional multiplication than traditional addition.

Are there any other compelling reasons?

Thank you!

Hey Battlemage!.

If you want a deeper reason for this you need to look at geometric algebra and consider the following problem:

You want to define an operation that for valid vector a and b, then given x = ab, then x/b = a for a given vector.

This is the kind of thing that Hermann Grassmann and later people like Hamilton, Clifford and other mathematicians were considering and nowadays, this kind of thing is at the heart of things like theoretical physics because it has an intuitive explanation and because it actually simplifies things dramatically.

What the consequence of this actually is, is that rotation is a natural way to interpret the characteristic of multiplication in this nature. The thing is extending this kind of property beyond complex numbers to quaternions (Hamilton), octonions (Cayley I think) and anything else. When you get to octonians you get non-associativity and things get crazy, but it is useful for theoretical physics for a number of reasons.

Because of this rotation characteristic, you can express things involving sine's and cosines', anything with rotations amongst other things in this geometric algebra. Also this is directly related to linear algebra as a general thing and cross products deal with this kind of 'vector multiplication' for three-dimensional vectors.

If you want a deeper knowledge take a look at any solid accounts on Geometric Algebra and if you want to see how it is used mathematically, look at any theoretical physics account that uses a geometric algebraic approach (you should find a few).

 

[algebrat]

For vectors A and B,

1. A dot B multiplies the length A of by the component of B parallel to A,
2. A cross B multiplies the length of A times the component of B perpendicular to A.

As somebody pointed, geometric algebra tries to take this into account simultaneously. Some applications, in math and physics, might find one or the other significant. Often, calculating work uses the dot product, while calculating torque uses the cross product.

I think the idea that addition was already used is a little overkill. This is not some barely understandable operation, it clearly looks like multiplication. The only problem is, there are two good versions, dot and cross, so we find notation for each, I suppose cross product is a good name for perpendicular multiplication. By the way, nobody has mentioned the wedge product, another multiplication to compare. If I remember correctly, it is a generalization of cross product to multiply forms on curved spaces of any dimension.

 

[mathwonk]

I have always considered, that in the presence of an addition, a "multiplication" is a binary operation which is distributive over the addition.

hence the rule (U+V)xW = UxW + VxW justifies calling the cross product a multiplication on vectors.