The University of Sydney - School of Mathematics and Statistics

Internal division of a line segment	Page 1 of 2

With an assigned point O as origin, the position of any point P is given uniquely by the vector , which is called the position vector of P relative to O.

Let P₁ and P₂ be any points, and let R be a point on the line P₁P₂ such that R divides the line segment P₁P₂ in the ratio m : n. That is, R is the point such that = . Our task is to find the position vector of R (relative to O) in terms of the position vectors of P₁ and P₂.

As = , we have n = m and therefore

(1)

which rearranges to give

When m and n are both positive, the vectors and have the same direction, since = . This corresponds to the situation where R lies between P₁ and P₂, as shown in the diagram above. R is then said to divide the line segment P₁P₂ internally in the ratio m : n.

As a special case of the general formula (1) we can obtain a formula for the position vector of the midpoint M between two points P₁ and P₂.

In this case, m : n = 1 : 1 and so