# Displacement vector for a cylinder

We have a cylinder of length $l$ (in units of its radius $d,$ as basic unit of length set to $d=1.$) in a box, and we consider an orthonormal Cartesian coordinate system with its origin placed at the centre of the box. We know the position of the cylinder in terms of the coordinates of its centre of mass, so $\vec{R}=(r_x,r_y,r_z)$ and in terms of its orientation vector (vector along its main/long axis) $\vec{O}=(o_x,o_y,o_z).$ Assuming at a later time the position and orientation of the cylinder have been randomized (still in the box), we'd like to estimate the displacement vector in terms of: how much the cylinder has moved parallel to its long axis ($\Delta \vec{R}_{||}$), how much it's moved in the plane perpendicular to its long axis ($\Delta \vec{R}_{\perp}$), how much it's been rotated.