# Origin of Vector Cross Product

I understand the cross product of vectors to some degree and i can calculate. But i don't really understand the origin of the cross product

What does a vector cross product mean in physical terms. Vector addition is quite easy to understand. I don't think the cross product is 'multiplication of vectors' as multiplication can be broken down into a series of additions at least for scalars.

Is the cross product simply a definition such that the product of 2 different components of a 2 vectors result in a multiplication of the magnitude in a direction perpendicular to the plane of the original vector components?

I don't really understand this in physical terms. All the books i have read have explained it in the above way.

mathman
The vector cross product, as well as the dot product and other vector manipulations, are all mathematical concepts. What do have in mind for "physical terms".

D H
Staff Emeritus
The origin of the vector cross product is the quaternion product. There was a minor skirmish in the field of mathematics at the end of the 19th century between the "quaternionists" (Hamilton et al) and the "vectorialists" (Gibbs, Heaviside, et al). Hamilton's quaternions were elegant, but maybe a bit too complex for everyday use -- and they did not quite fit into our three dimensional universe. Vectors are not quite as mathematically elegant, but they are simpler, and at least on the surface, they fit our 3D universe to a T.

Is the cross product simply a definition such that the product of 2 different components of a 2 vectors result in a multiplication of the magnitude in a direction perpendicular to the plane of the original vector components?

Sometimes, yes, that is precisely the way it's defined. In others, we just define it using the vector notation. There are many more ways to define it. In 3 dimensions, I'm pretty sure most of them are equal, at least for most vectors.

D H said:
The origin of the vector cross product is the quaternion product.
That's not entirely true. It was studied as a useful geometric construct much earlier. You have to remember that geometry is very old, and quaternions are a relatively recent concept.

In the end, what's important is not where it came from but the fact that it stuck around, mainly because it's so useful.

How is it the cross product so much useful in "electrodynamics" was discovered before it.
Is it a mere coincidence that laws of electrodynamics can so easily be expressed using the cross product

Char. Limit
Gold Member
Nothing is coincidence...

Torque also uses cross product.

$$\vec{T}=\vec{F} \times \vec{r}$$

Torque is:

$$\vec{T}=\vec{r} \times \vec{F}$$

Cross product is noncommutative.

Char. Limit
Gold Member
Ah...

Thanks, I've always read it as T=Fr for perpendicular forces and radii, and never knew that it was different.

If I remember anything from my multivariable calc, it can be expressed as $$\vec{T}=-\vec{F} \times \vec{r}$$ if you would like. I've usually seen it in r cross F though.