End-to-end vector

In the physical chemistry study of polymers, the end-to-end vector is the vector that points from one end of a polymer to the other end.

If each monomer unit in a polymer is represented by a point in space, the translation vectors ${\displaystyle {\vec {r}}_{i}}$ connect between these points. The end-to-end vector ${\displaystyle {\vec {R}}}$ is the sum of these translation vectors:

${\displaystyle {\vec {R}}=\sum _{i}{\vec {r}}_{i}}$

The norm of the end-to-end vector is called the end-to-end distance.

The quadratic mean of the end-to-end distance ${\displaystyle \left\langle r^{2}\right\rangle }$ can be related to the quadratic mean of the radius of gyration ${\displaystyle \left\langle s^{2}\right\rangle }$ of a polymer by the relation:¹

${\displaystyle \left\langle r^{2}\right\rangle =6\left\langle s^{2}\right\rangle }$