# Angular Momentum, Skew Symmetry and Cross Product

I am able to justify how the cross product between the postition from an axis and the velocity of a particle, provides "angular velocity." Such an operation maintains "information" about the instantaneous axis of rotation, the magnitudes of the perpedicular disance from the axis and the velocity and mass.

I also know that we can obtain the same result by taking the position vector, converting its components into a skew symmetric matrix and multiplying that matrix by the linear momentum vector.

Good.

What I am UNABLE to do (and the reason I solicit guidance) is to explain how a skew symmetric matrix of position components times the momentum vector gives the angular momentum.

In other words, it seems I am still tied to the cross product.

How can I "explain" the meaning of the angular momentum WITHOUT resorting to the cross product and going DIRECTLY to the skew symmetric form?

• You shouldn't consider the vector notation. Consider the index notation and write the components, then you will see what the matrix actually does. And, not surprisingly, what it does is exactly the same as what the vector product with $x$ does. So, there is no new intuition behind the matrix vector product... – Photon May 10 '17 at 8:27