Vectors $\vec{A}$ and $\vec{B}$ are equal if they have the same magnitude and direction. Is there a mathematical relation defined to apply to them only if they also have the same origin?
Actually, I would like to have concepts to distinguish between these three cases:
(I) $\vec{A}$ is equal to $\vec{B}$ but with different origin,
(II) $\vec{A}$ is equal to $\vec{B}$ with the same origin,
(III) $\vec{A}$ is the same entity as $\vec{B}$.
As a physical example, the difference between (II) and (III) could be
(II - example) $\vec{F}_1$ and $\vec{F}_2$ are two forces applied on the same point,
(III - example) $\vec{F}_1$ and $\vec{F}_2$ are actually a single force, the same entity.