Vector Products: What Happens When Professors Refuse to Explain?

[Treadstone 71]

Why is a vector product not generally defined pointwise? My professor simply gave a mysterious "you'll see why later". What's the worst that can happen?

 

[matt grime]

What do you mean 'pointwise'? And for that matter what do you mean by 'a vector product', there is generally considered to be one such. Presumably he means something like 'because there is no nice formula for what v^w is in terms of v and w', though I'm not sure I agree with that. Maybe if you even gave what your definition is we might have some better idea.

 

[Treadstone 71]

By "pointwise" I mean "component-wise". Define x:VXV->V as (a,b,c,d)x(e,f,g,h)=(ae,bf,cg,dh) to be the vector multiplication on some vector space V.

 

[matt grime]

Oh, right, so that's what you mean. Since vector product has a strict meaning wondered what you were up to, as does pointwise which is strictly different from what you've done: it literally is used to talk about things happening at one point at a time.

Well you can define a 'multiplication' like that, it is certainly a bilinear map from VxV to V that does a lot of interseting things, it even has a name that I can't recall (when applied to a vector space of matrices), and it makes V into an algebra. But it is however, at the level you're looking at useless: it does nothing geometric, or nice.

 

[masudr]

As matt grime has pointed out, there is at most levels no need to introduce this kind of a product.

Instead we have scalar and vector products defined (on \mathbb{R}^3) as the products of magnitudes, the cosine/sine of the angle between them (and we make the vector product into a vector by the right-hand screw rule).

Then we can work out what to do with the components once we've got a definite basis. And neither of these correspond to the operation you've defined.