5. Division of a line segment
 Glossary Examples Advanced Examples

### Internal division of a line segment

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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

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