Suppose A represents a unit vector directed along the axis of...

For a purely rotating body, the velocity $\vec{v}$ of any particle $P$ situated away from the axis of rotation is given by $\vec{v} = \vec{\omega} \times \vec{r}$, where $\vec{\omega}$ is the angular velocity vector and $\vec{r}$ is the position vector of the particle.

The angular velocity vector $\vec{\omega}$ is directed along the axis of rotation.

From the properties of the cross product, the velocity vector $\vec{v}$ is perpendicular to both $\vec{\omega}$ and $\vec{r}$.

Since $\vec{A}$ is a unit vector along the axis of rotation (parallel to $\vec{\omega}$)$and \vec{B}$ is a unit vector along the velocity $\vec{v}$, $\vec{A}$ is perpendicular to $\vec{B}$.

$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(90^{{}^{{}^{\circ}}}) = 0$