# Derivation regarding spatial velocity

In the rigid body dynamics algorithm book (Featherstone et al.) it claims

We start with a rigid body, $$B$$, and choose a fixed point, $$O$$, which can be located anywhere in space. (See Figure 2.1(a).) Given $$O$$, the velocity of $$B$$ can be specified by a pair of 3D vectors: the linear velocity, $$v_O$$, of the body-fixed point that currently coincides with $$O$$, and an angular velocity vector, $$\omega$$. The body as a whole can then be regarded as translating with a linear velocity of $$v_O$$, while simultaneously rotating with an angular velocity of $$\omega$$ about an axis passing through $$O$$. From this description, we can calculate the velocity of any other point in the body using the formula $$v_p = v_O + \omega \times \vec{OP}$$ where $$P$$ is the point of interest, $$v_P$$ is the velocity of the body-fixed point currently at P, and $$\vec{OP}$$ gives the position of $$P$$ relative to $$O$$.

A few things I am confused with is

• What is $$v_O$$, from the description it seems like $$v_O$$ is not the same as the linear velocity of the body frame in the world frame.
• $$\omega$$ is described as the angular velocity about an axis passing through $$O$$, so it seems like $$\omega$$ is neither the body nor the spatial angular velocity, but something different.

I do not understand how to derive this result, can someone help me with this?