A body rotates uniformly about an axis fixed in an inertial frame...

Since the body rotates uniformly about a fixed axis, its angular acceleration is zero.

The particle $P$ moves in a circular path centered on the axis of rotation. The only acceleration of the particle is the centripetal acceleration, which is directed towards the center of its circular path.

This means the resultant force $\vec{F}$ acting on the particle is directed towards the axis and is perpendicular to the axis of rotation.

Since $\vec{A}$ is along the axis of rotation and $\vec{B}$ is along the resultant force, the angle between them is $90^{{}^{{}^{\circ}}}$.

Therefore, $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos 90^{{}^{{}^{\circ}}} = 0$.