Section‌ ‌Formula‌ ‌in‌ ‌3‌ ‌Dimension‌

Section formula is applied in the three-dimensional geometry to find the coordinates of points dividing a line segment. This division of a line segment occurs internally and in a specific ratio. Section formula uses coordinate geometry to find the coordinates of these points in space.

The three-dimensional section formula uses a three-coordinate system to denote the coordinates of points. In the same way, the section formula is applicable in two-dimensional geometry as well, but in a two-coordinate system.

Section formula

Consider two points A (x1, y1, z1) and B (x2, y2, z2). Consider another point P on the line AB with coordinates (x, y, z) that divides the line in m:n ratio.

 1. We need to draw perpendiculars AL, PN, and BM to the XY plane such that AL || PN || BM.
 2. The points M, N and L lie on the straight line formed due to the intersection of AL, PN and BM on the XY plane.
 3. A line ST is drawn from point P such that ST is parallel to LM.
 4. The line ST intersects BM internally at T and AL at S.
 5. Since ST is parallel to LM and PN || AL || BM ||, therefore, these lines form two parallelograms, LNPS and NMTP.

We can represent external section formula as:

Example: What are the coordinates of the line segment (1, -2, 3) and (3, 4, -5), which divide the line segment in the ratio of 2:3 internally and externally?

Solution:

We know the internal section formula is given by:

We will get the coordinates of P as: P = (-3, -14, 19)