Section Formula

Concept: The Angle Bisector Theorem states that the internal angle bisector of a triangle divides the opposite side in the ratio of the adjacent sides. If the angle bisector from meets at , then Thus, point divides the line segment internally in the ratio . If a point divides the line joining vectors and in the ratio , then its position vector is
Step 1: {Apply the Angle Bisector Theorem.} Since point divides internally in the ratio
Step 2: {Use the section formula in vector form.} If a point divides the segment joining and in the ratio , its position vector is Rearranging,
Step 3: {Write the final position vector.} Thus the position vector of is