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