A man needs to hang two lanterns on a straight wire whose end points have coordinates $\displaystyle \mathrm{A}(4,1,-2)$ and $\displaystyle \mathrm{B}(6,2,-3)$. Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB .

Let $\displaystyle P$ and $\displaystyle Q$ trisect the wire $\displaystyle \mathrm{A} \mathrm{B}$. $\displaystyle P$ divides $\displaystyle \mathrm{A} \mathrm{B}$ in the ratio $\displaystyle 1$:$\displaystyle 2$ then, coordinate of point $\displaystyle P=\left(\frac{14}{3}, \frac{4}{3},-\frac{7}{3}\right)$ $\displaystyle Q$ divides $\displaystyle \mathrm{A} \mathrm{B}$ in the ratio $\displaystyle 2$:$\displaystyle 1$ then, coordinate of point $\displaystyle Q=\left(\frac{16}{3}, \frac{5}{3},-\frac{8}{3}\right)$