I have the following equation:
$ F_{i j}^{e}(x)=k_{i j}\left(\left\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|-l_{i j}\right) \frac{\mathbf{x}_{i}-\mathbf{x}_{j}}{\left\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|} $
from this paper http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Magnenat-Thalmann04.pdf on cloth simulation, it occurs on page 19. I just wanted to clarify what the meaning behind this equation would be.
So I believe it is saying that the Force between particles i and j is equal to the k constant multiplied by $ \left\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|-l_{i j} $ , which I would assume to be the distance (magnitude) between two particles, i and j, take away the rest length. But then in the fraction I can see that there is also $ \mathbf{x}_{i}-\mathbf{x}_{j} $, so without the ||, and I can't tell the difference in meaning between $ \mathbf{x}_{i}-\mathbf{x}_{j} $ and $ ||\mathbf{x}_{i}-\mathbf{x}_{j} || $ . I've looked for the meaning of || || in mathematics, but it appears to have many meanings, including the norm but in this context I am unsure as I haven't cowered norms in the limited university maths modules (I had one) and I don't really know how it applies here after looking for the definition. Could someone clarify?
This looks like a force, and typically the force is not just a real number, it also usually has a direction. That's what the $x_i-x_j$ is contributing. All the other quantities appearing combined to form a coefficient in front of the vector $x_i-x_j$.
I can't see anything to suggest that $\|\cdot\|$ means anything other than the Euclidean norm on real vectors.