- #1

stoopkid

- 6

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[itex]\vec{B}[/itex] = [itex]\frac{\mu_{0}}{4\pi}[/itex][itex]\frac{\vec{v}×\left(\vec{r}-\vec{r_{0}}\right)}{\left\|\vec{r}-\vec{r_{0}}\right\|^{3}}[/itex]

So I took the curl and ended up with the following:

∇×[itex]\vec{B}[/itex] = [itex]\frac{1}{\left\|\vec{Δr}\right\|^{5}}[/itex][itex]\left[\stackrel{\stackrel{\Large\left(3Δx\vec{v}-v_{x}\vec{Δr}\right)\bullet\vec{Δr}}{\normalsize\left(3Δy\vec{v}-v_{y}\vec{Δr}\right)\bullet\vec{Δr}}}{\scriptsize\left(3Δz\vec{v}-v_{z}\vec{Δr}\right)\bullet\vec{Δr}}\right][/itex]

where x,y,z are components of [itex]\vec{r}[/itex], and Δx = (x-x[itex]_{0}[/itex]), and [itex]\vec{Δr} = \vec{r} - \vec{r_{0}}[/itex].

So I was wondering if there is an operation which takes a vector A, and scalar multiplies each of its components to a vector B, and the resulting vectors become components of a new vector. For example, the x component of the new vector would be: a[itex]_{x}\vec{B}[/itex], so it would be a "vector of vectors". I was also wondering if there is an operation which would take a vector of vectors [itex]\vec{C}[/itex] and dot products each of its components by another vector, [itex]\vec{R}[/itex], and the resulting scalars from those operations form the new components, i.e. the x-component of the new vector would be: [itex]\vec{C_{x}}\bullet\vec{R}[/itex].

If we call the first operation [itex]\otimes[/itex], and the second operation [itex]\odot[/itex], then we can rewrite the curl equation:

[itex]∇×\vec{B}=\frac{\left(3\vec{Δr}\otimes\vec{v}-\vec{v}\otimes\vec{Δr}\right)\odot\vec{Δr}}{\left\|\vec{Δr}\right\|^{5}}[/itex]

Both the operations are bilinear and non-commutative. I was wondering if there is a way to generalize the idea of this:

The scalar product [itex]a\vec{v} = \left(a*v_{1}, a*v_{2}, a*v_{3}\right)[/itex], takes the scalar a as a whole distributes it over the components of [itex]\vec{v}[/itex], and puts each of these products into the component of a new vector. What about a scalar product that instead adds these up? I.e. [itex]a\vec{v} = a*v_{1}+a*v_{2}+a*v_{3}[/itex].

The dot product [itex]\vec{a}\bullet\vec{v} = a_{1}*v_{1} + a_{2}*v_{2} + a_{3}*v_{3}[/itex] multiplies corresponding components of the two vectors and adds these all up. What about a dot product that instead of adding these into a vector? I.e. [itex]\vec{a}\bullet\vec{v} = (a_{1}*v_{1}, a_{2}*v_{2}, a_{3}*v_{3})[/itex]

Or what about an operation where a vector distributes like a scalar over the components of another vector? This is the operation [itex]\otimes[/itex] that came up in my original problem, where [itex]\vec{a}\otimes\vec{v} = (a_{1}\vec{v}, a_{2}\vec{v}, a_{3}\vec{v})[/itex], which is a vector of vectors. What about a case where instead of being components of a vector, the components were all added up?

Is there a way in which these are all specific examples of a general "kind of operation", where we can specify how the items being multiplied distribute over each other's components? I know linear algebra covers some of this stuff, but I don't remember learning anything about "vectors of vectors".