# Why are there two definitions for the angular momentum of a particle with respect to a moving point?

There seem to be two definitions for the angular momentum of a particle with respect to a moving point.

See the figure below. Let $$O$$ be the origin of a coordinate system in an inertial frame. Let $$Q$$ be a point moving with respect to $$O$$. Let $$K$$ be the position of the particle. $$\vec r_Q$$ is the vector from $$O$$ to $$Q$$, $$\vec r_K$$ is the vector from $$O$$ to $$K$$, and $$\vec r_{KQ} = \vec r_K - \vec r_Q$$ is the vector from $$Q$$ to $$K$$. The velocity of $$K$$ in the inertial frame is $$\vec v_K$$, the velocity of Q in the inertial frame is $$\vec v_Q$$, and the velocity of $$K$$ with respect to $$Q$$ is $$\vec v_ {KQ} = \vec v_ K - \vec v_Q$$. Some physics texts seem to define the angular momentum of $$K$$ with respect to $$Q$$, $$\vec L_{KQ}$$, as $$(1) \vec L_{KQ} = m[(\vec r_K - \vec r_Q) \times (\vec v_K - \vec v_Q)]$$ For example, see Symon, Mechanics

Engineering texts seem to use the definition $$(2) \vec L_{KQ} = m[(\vec r_K - \vec r_Q) \times \vec v_K]$$ For example, see the following text by Kochmann online: https://ethz.ch/content/dam/ethz/special-interest/mavt/mechanical-systems/mm-dam/documents/Notes/Dynamics_LectureNotes.pdf.

Updated Information

I recently found an MIT physics class lecture online that uses relationship (2). See https://www.youtube.com/watch?v=NHedXxUO-Bg. The professor specifically emphasizes that for $$\vec L_{KQ} = (\vec r_K - \vec r_Q) \times m\vec p$$ where $$\vec p$$ is the linear momentum of the particle, the momentum must be evaluated using the velocity of $$K$$ with respect to $$O$$, that is $$\vec p = m\vec v_K$$.

Is relationship (1) ever used in practice?