I was thinking, if exist a product (cross) between vectors defined as:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\vec{a}\times\vec{b}=a\;b\;sin(\theta)\;\hat{c}[/tex]

and a product (dot) such that:

[tex]\vec{a}\cdot\vec{b}=a\;b\;cos(\theta)[/tex]

Why not define more 2 products that result:

[tex]\\a\;b\;sin(\theta) \\a\;b\;cos(\theta)\;\hat{d}[/tex]

So, for my proper use and consume, I thought to create the follows definitions:

[tex]\begin{matrix} \cdot & \hat{i} & \hat{j} & \hat{k} \\ \hat{i} & 1 & 0 & 0 \\ \hat{j} & 0 & 1 & 0 \\ \hat{k} & 0 & 0 & 1 \\ \end{matrix}[/tex][tex]\begin{matrix} \times & \hat{i} & \hat{j} & \hat{k} \\ \hat{i} & 0 & 1 & -1 \\ \hat{j} & -1 & 0 & 1 \\ \hat{k} & 1 & -1 & 0 \\ \end{matrix}[/tex][tex]\begin{matrix} \ \odot & \hat{i} & \hat{j} & \hat{k} \\ \hat{i} & \hat{i} & \vec{0} & \vec{0} \\ \hat{j} & \vec{0} & \hat{j} & \vec{0} \\ \hat{k} & \vec{0} & \vec{0} & \hat{k} \\ \end{matrix}[/tex][tex]\begin{matrix} \ \otimes & \hat{i} & \hat{j} & \hat{k} \\ \hat{i} & \vec{0} & \hat{k} & -\hat{j} \\ \hat{j} & -\hat{k} & \vec{0} & \hat{i} \\ \hat{k} & \hat{j} & -\hat{i} & \vec{0} \\ \end{matrix}[/tex]

I think that this definitions to generate new possibilities and facilitate some notations. For example:

[tex]\frac{\partial^2 f}{\partial x^2}\frac{dx^2}{dt^2}+\frac{\partial^2 f}{\partial y^2}\frac{dy^2}{dt^2}=\bigtriangledown^2f\cdot \frac{d\vec{r}}{dt}\odot \frac{d\vec{r}}{dt}[/tex]

This is only a ideia that I'd like to share, is not a doubt. What do you think? It seems useful and applicable?

BTW, this definitions extends and generates some interesting questions:

If I can apply a scalar field f in:

[tex]\frac{\partial }{\partial x}\hat{x}+\frac{\partial }{\partial y}\hat{y}[/tex]

Can I apply a scalar field f in this version of Del operator too:

[tex]\left ( \frac{\partial }{\partial x}-\frac{\partial }{\partial y}\right )dxdy\;\hat{k}[/tex]

?

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# An extension of Dot and Cross Product

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