If the angular momentum of a body is zero at some point, is it necessary that it will be zero at a different point ?
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The definition of angular momentum for a point particle is a pseudovector r×p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector p = mv. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it.