mz nz z m Similarly, by drawing perpendiculars to the XZ and YZ-planes, we get my +ny mx +nx y= and x= m+n m+n Hence, the coordinates of the point R which divides the line segment joining two points P ( x , y , z ) and Q ( x , y , z ) internally in the ratio m : n are m nz mz m ny my m nx mx , , If the point R divides PQ externally in the ratio m : n , then its coordinates are obtained by replacing n by – n so that coordinates of point R will be m nz mz m ny my m nx mx , , Case Coordinates of the mid-point: In case R is the mid-point of PQ, then m : n = : so that x = and , z z z These are the coordinates of the mid point of the segment joining P ( x , y , z ) and Q ( x , y , z ). Fig . INTRODUCTION TO THREE DIMENSIONAL GEOMETRY Case The coordinates of the point R which divides PQ in the ratio k : are obtained by taking m k which are as given below: k z kz k ky k k , , Generally, this result is used in solving problems involving a general point on the line passing through two given points. Example Find the coordinates of the point which divides the line segment joining the points ( , – , ) and ( , , – ) in the ratio : (i) internally, and (ii) externally.
Solution (i) Let P ( x , y , z ) be the point which divides line segment joining A( , – , ) and B ( , , – ) internally in the ratio : . Therefore ( ) + ( ) + , ( ) + (– ) + , (– )