Question regarding cross products and determinants

[Salamon]

I understand how to calculate a vector cross product. I also understand that in order to calculate a vector cross product we are calculating the determinant of a 3 x 3 matrix.

I guess I have a few questions.

1)How did mathematicians or physicists discover how to calculate the determinant of a 3 x 3 matrix, and what does this determinant represent?


For example, A x B = (AyBz - ByAz)i - (AxBz-BxAz)j + (AxBy-BxAy)k.

2) How did we know to put a negative in front of the j term?

3) How did we recognize that phenomena like Torque and Magnetic Force could be described by cross products?

I hope that I am being clear enough and don't sound like a confused idiot. I'd appreciate answers to any of these questions.

 

[chiro]

I understand how to calculate a vector cross product. I also understand that in order to calculate a vector cross product we are calculating the determinant of a 3 x 3 matrix.

I guess I have a few questions.

1)How did mathematicians or physicists discover how to calculate the determinant of a 3 x 3 matrix, and what does this determinant represent?


For example, A x B = (AyBz - ByAz)i - (AxBz-BxAz)j + (AxBy-BxAy)k.

2) How did we know to put a negative in front of the j term?

3) How did we recognize that phenomena like Torque and Magnetic Force could be described by cross products?

I hope that I am being clear enough and don't sound like a confused idiot. I'd appreciate answers to any of these questions.

Hey Salamon and welcome to the forums.

There is a deep history to the subject that is known as geometric algebra.

Lots of different people were considering different kinds of algebras on multi-dimensional vectors (i.e. not scalars) and how to think about them in some kind of unified way.

A german guy known as Hermann Grassmann was considering the idea of multiplication and division of vectors in such a way that if you did (a*b)/b = a you would get something that made sense and fitted those identities.

From that idea you got what is known as a geometric product and from that you get two main quantities known as the interior (or inner) product and the exterior (or outer) product and both of these form a bi-vector describing the product of two vectors (just like a complex number has two independent quantities for the real and imaginary parts).

So in a sense, this idea of being able to multiply and divide vectors helped formulate a lot of the base ideas for the framework of geometric algebra, but only for 3-dimensions but also many dimensions where you could do this.

Now the cross-product does satisfy the above where you can take A X B and then multiply it by a vector corresponding to 1/B to get back A, but it's not the only algebra that allows this: the quaternions are also an example of a division algebra that use a four-dimensional object instead of a three dimensional one and work exactly the same as a 3D vector algebra (if you omit the special scalar term and set it to 0, you'll get your vector term to be the cross product when you multiply the two quaternions).

Now the generalized linear object is just a matrix, so when you consider general results of vector and geometric algebras that have some property of linearity, you are going to get results that deal with matrices and properties of matrices like determinants.

So that should aid you for the first two.

For the third question, one must understand that orientation provides a way to analyze rotation in a general sense and orientation comes about from looking at the vector product and its algebra.

If you look at the development of vector algebra (especially the stuff done by a guy named David Hestenes who introduced a lot of this stuff into physics), you'll find that when you take what Grassmann did and analyze what happens in terms of the bi-vector (remember the inner and outer products: outer product is just cross product and inner product is dot product in three dimensions), you see that there is a rotation going on in the same kind of way that you get when you multiply two complex numbers.

This rotation element is a general feature of these kinds of vector algebras and when considering rotations in 3D space, it is not surprising that the algebra used to describe such systems ends up involving cross-products, quaternions or some high level division algebra (division algebra just means you can do division which is an inverse of multiplication).

 

[Salamon]

Thank you. That helped me a lot.