- #1

Trying2Learn

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- TL;DR Summary
- The question says it all.

So I do know that there does exist a generalization of the cross product (the exterior product), but this question does not concern that (and I would prefer it not )

I know that the cross product (that Theodore Frankel, for example, calls "the most toxic operation in math") works in 3D only. (Why does he say this? Simply because it fails associativity?)

I am aware that the operation has two vectors as input, takes their magnitude and the angle between them and outputs a vector that is perpendicular to both and contains information about the angle and the magnitudes.

I can REASON out why this strange operation is so useful when constructing the "moment." I can reason out why it can deliver information about the "tendency" to keep rotating (angular momentum) and can work the operations to show that the rate of change of angular momentum is the moment.

All that is fine.

However, despite that, this operation unnerves me, and I do not know why.

I can see (in my mind) how integrals "sum up effects." I can see in my mind, the role of differentiating. I can see the role of the dot product (in 3D space, not its generalizations). However, this operation called the "cross product" seems to be like rabbit pulled out of a hat: "Oh! It's useful! So, let's use it."

Can anyone discuss the role/need/history of this operation. It seems laden with baggage (i.e.: the "right hand rule" to determine the direction of the resulting vector)

Why (how?) did this operation come about? It just seems whimsical (esp. the sine of the angle between the vectors and, say, not the cosine)

Anything?

I know that the cross product (that Theodore Frankel, for example, calls "the most toxic operation in math") works in 3D only. (Why does he say this? Simply because it fails associativity?)

I am aware that the operation has two vectors as input, takes their magnitude and the angle between them and outputs a vector that is perpendicular to both and contains information about the angle and the magnitudes.

I can REASON out why this strange operation is so useful when constructing the "moment." I can reason out why it can deliver information about the "tendency" to keep rotating (angular momentum) and can work the operations to show that the rate of change of angular momentum is the moment.

All that is fine.

However, despite that, this operation unnerves me, and I do not know why.

I can see (in my mind) how integrals "sum up effects." I can see in my mind, the role of differentiating. I can see the role of the dot product (in 3D space, not its generalizations). However, this operation called the "cross product" seems to be like rabbit pulled out of a hat: "Oh! It's useful! So, let's use it."

Can anyone discuss the role/need/history of this operation. It seems laden with baggage (i.e.: the "right hand rule" to determine the direction of the resulting vector)

Why (how?) did this operation come about? It just seems whimsical (esp. the sine of the angle between the vectors and, say, not the cosine)

Anything?