Section Formula
Let the two given points be P ( x1 , y1 , z1 ) and Q ( x2 , y2 , z2 ). Let the point R ( x , y , z ) divide PQ in the given ratio m : n internally . Coordinates of Point R will be ,
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
Special case :
Coordinates of the mid-point : In case R is the mid-point of PQ , then m : n = 1 : 1 . The coordinates of point R which divides PQ in ratio k : 1 are obtained by taking k = m / n
Numerical : Find coordinates of point which divides line segment joining points ( 1 , – 2 , 3 ) and ( 3 , 4 , – 5 ) in the ratio 2 : 3 internally , and externally .
Solution : Here m = 2 & n = 3 . Coordinates of Point P when it divided m : n internally is
Px = ( mx2 + nx1 ) / ( m + n ) = ( 2 * 3 + 3 * 1 )/( 2 + 3 ) = 9 / 5