 Section Formula in 3D

In two dimension geometry, the concept of section formula is implemented to find the coordinates of a point dividing a line segment internally in a specific ratio. In order to locate the position of a point in space, we require a coordinate system. After choosing a fixed coordinate system in three dimensions, the coordinates of any point P in that system can be given. In case of a rectangular coordinate system, it is given by an ordered 3-tuple (x, y, z). Also, if the coordinates (x, y, z) are already known then we can easily locate the point P in space. The concept of section formula can be extended to three dimension geometry as well as to determine the coordinates of a point dividing a line in a certain ratio.

Section Formula

Let us consider two points A (x1, y1, z1) and B(x2, y2, z2). Consider a point P(x, y, z) dividing AB in the ratio m:n as shown in the figure given below. To determine the coordinates of the point P, the following steps are followed: • Draw AL, PN, and BM perpendicular to XY plane such that AL || PN || BM as shown above.
• The points L, M and N lie on the straight line formed due to the intersection of a plane containing AL, PN and BM and XY- plane.
• From the point P, a line segment ST is drawn such that it is parallel to LM.
• ST intersects AL externally at S and it intersects BM at T internally.

Since ST is parallel to LM and AL || PN || BM, therefore, the quadrilaterals LNPS and NMTP qualify as parallelograms.

Also, ∆ASP ~∆BTP therefore,

$\large \frac{m}{n} = \frac{AP}{BP} = \frac{AS}{BT} = \frac{SL – AL}{BM – TM} = \frac{NP – AL}{BM – PN} = \frac{z – z_{1}}{z_{2} – z}$

Rearranging the above equation we get,

$\large \frac{mz_{2}+ nz_{1}}{m + n}$

The above procedure can be repeated by drawing perpendiculars to XZ and YZ- planes to get the x and y coordinates of the point P that divides the line segment AB in the ratio m:n internally.

$\large x = \frac{mx_{2}+ nx_{1}}{m+n}, y = \frac{my_{2}+ ny_{1}}{m+n}$

Sectional Formula (Internally):

Thus, the coordinates of the point P(x, y, z) dividing the line segment joining the points A(x1, y1, z1) and B(x2, y2, z2) in the ratio m:n internally are given by:

$\large \left ( \frac{mx_{2}+ nx_{1}}{m+n}, \frac{my_{2}+ ny_{1}}{m+n}, \frac{mz_{2}+ nz_{1}}{m+n} \right )$

Sectional Formula (Externally):

If the given point P divides the line segment joining the points A(x1, y1, z1) and B(x2, y2, z2) externally in the ratio m:n, then the coordinates of P are given by replacing n with –n as:

$\large \left ( \frac{mx_{2}- nx_{1}}{m+n}, \frac{my_{2}- ny_{1}}{m+n}, \frac{mz_{2}- nz_{1}}{m+n} \right )$

This represents the section formula for three dimension geometry.

If the point P divides the line segment joining points A and B internally in the ratio k:1, then the coordinates of point P will be

$\large \left ( \frac{kx_{2}+ x_{1}}{k+1}, \frac{ky_{2}+ y_{1}}{k+1}, \frac{kz_{2}+ z_{1}}{k+1} \right )$

What if the point P dividing the line segment joining points A(x1, y1, z1) and B(x2, y2, z2) is the midpoint of line segment AB?

In that case, if P is the midpoint, then P divides the line segment AB in the ratio 1:1, i.e. m=n=1.

Coordinates of point P will be given as:

$\large \left ( \frac{1 \times x_{2}+ 1 \times x_{1}}{1+1}, \frac{1 \times y_{2}+ 1 \times y_{1}}{1+1}, \frac{1 \times z_{2}+ 1 \times z_{1}}{1+1} \right )$

Therefore, the coordinates of the midpoint of line segment joining points A(x1, y1, z1) and B(x2, y2, z2) are given by,

$\large \left ( \frac{x_{2}+ x_{1}}{2}, \frac{y_{2}+ y_{1}}{2}, \frac{z_{2}+ z_{1}}{2} \right )$<