Feature Request: Left (and Right) inverse of the Cross Product

Leon Deligny via NumPy-Discussion <[email protected]>
Newsgroups gmane.comp.python.numeric.general
Message-ID <[email protected]>
### Proposed new feature or change:

Motivations: This is specific to 3D vector algebra. In Fluid Dynamics, we have access to the moment of a force at a specific point (M_P = OP \cross F). This calculation is crucial when determining the center of pressure (CoP), a pivotal concept for understanding the distribution of forces on an object submerged in a fluid. To accurately pinpoint the CoP, we often need to "reverse" this process, effectively requiring an inverse functionality for the cross product.

How to compute the right-inverse: Given two real numbers a and c we want to find b such that c = a \cross b . If we write this equation in matrix format then we need to define:

```latex
A = \begin{pmatrix}
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0
\end{pmatrix}
```

to get $c = A \cdot b$ (where $\cdot$ is the matrix multiplication). In real case scenarios there does not exist a vector b such that $c = A \cdot b$. So we can always find a vector b such that $|c - A \cdot b|_2^2$ is minimal (i.e. b is the best approximation $c \approx A \cdot b$).

When minimizing $|c - A \cdot b|_2^2$ there exists an analytical solution found with gaussian elimination procedure (we are working with 3x3 matrices and 3-dimensional vectors).

I did not find any litterature on this subject specifically. But this would be a greatly appreciated feature.
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