5. Division of a line segment





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 P_{1} and P_{2} be any points, and let R be a point on the line P_{1}P_{2} such that R divides the line segment P_{1}P_{2} 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 P_{1} and P_{2}. As = , we have n = m and therefore
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 P_{1} and P_{2}, as shown in the diagram above. R is then said to divide the line segment P_{1}P_{2} 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 P_{1} and P_{2}. In this case, m : n = 1 : 1 and so


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