+ ( ) – + z Thus, the required point is , , (ii) Let P ( x , y , z ) be the point which divides segment joining A ( , – , ) and B ( , , – ) externally in the ratio : . Then ( ) + (– )( ) = – , + (– ) ( ) + (– )(– ) = – + (– ) , (– ) + (– )( ) = + (– ) z Therefore, the required point is (– , – , ). Example Using section formula, prove that the three points (– , , ), ( , , ) and ( , , – ) are collinear. Solution Let A (– , , ), B ( , , ) and C( , , – ) be the given points.
Let the point P divides AB in the ratio k : . Then coordinates of the point P are , , k k k k k k Let us examine whether for some value of k , the point P coincides with point C. On putting – = + k k , we get k = − MATHEMATICS When k = − , then ( k k and ( k k = − Therefore, C ( , , – ) is a point which divides AB externally in the ratio : and is same as P.Hence A, B, C are collinear. Example Find the coordinates of the centroid of the triangle whose vertices are ( x , y , z ), ( x , y , z ) and ( x , y , z ).
Solution Let ABC be the triangle. Let the coordinates of the vertices A, B,C be ( x , y , z ), ( x , y , z ) and ( x , y , z ), respectively. Let D be the mid-point of BC. Hence coordinates of D are , , z z Let G be the centroid