 # Section Formula in 3 Dimension

### Section Formula in 3D

While studying 2-dimensional geometry, you would have come across concepts of section formula. A line segment is divided internally in a specific ratio with the help of a point. In order to find the coordinates of a point, you apply the section formula. The coordinates of a point in the three-dimensional space are used to locate the point P that is given in the system. An ordered 3-tuple (x, y, z) is given to a coordinate system in case of a rectangle. In case the coordinations are already known, then it is easier to locate the point in the space. This concept of section formula can also be applied to the 3 dimension geometry and it can help determine the point which divides a line internally in a particular ration.

### Section Formula in Three Dimension Geometry

Consider two points P ( x1, y1, z1) and Q ( x2, y2, z2 ). Now PQ is divided in a ration m : n and the point dividing it is considered as M ( x, y, z ). The image of the same is given below.

The following steps are followed to determine the coordinates of the point M.

Steps to be followed for the derivation of section formula in 3d geometry are:

• Draw PU, MV, and QW perpendicular to XY plane such that PU || MV || QW as shown above.

• The points U, V, and W lie on the straight line formed due to the intersection of a plane containing PU, MV, and QW and XY- plane.

• From point M, a line segment KL is drawn such that it is parallel to UW.

• KL intersects PU externally at K and it intersects QW at L internally.

Since KL is parallel to UW and PU || MV || QW, therefore, the quadrilaterals UVMP and VWLM qualify as parallelograms.

Also, ∆PKM ~∆QLM therefore,

$\frac{m}{n}$ = $\frac{PM}{QM}$ = $\frac{PK}{QL}$ = $\frac{KU-PU}{QW-LW}$ = $\frac{VM-PU}{QW-MV}$

$\frac{m}{n}$ = $\frac{z-z_{1}}{z_{2}-z}$

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

By drawing a perpendicular to the XZ plane and the YZ plane to find the coordinate x and coordinate y of point M divides the line PQ internally int the ratio m:n internally.,  you can repeat the above-stated procedure.

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

### Sectional Formula ( Internally )

The coordinates of a point M ( x, y, z ) divides a line joining the two points P ( x1, y1, z1) and Q ( x2, y2, z2 ) in the ratio m:n internally and they go by:

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

### Sectional Formula ( Externally )

The coordinates of a point M ( x, y, z ) divides a line joining the two points P ( x1, y1, z1) and Q ( x2, y2, z2 ) in the ratio m : n externally and they go by:

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

The above representation of the section formula in 3-dimensional geometry can be shown as above.

If midpoint M divides the line segment, then the two points P ( x1, y1, z1) and Q ( x2, y2, z2 ) are in the ratio k : 1 internally and they go by:

$(\frac{kx_{2}+x_{1}}{m+n}$, $\frac{ky_{2}+y_{1}}{m+n}$, $\frac{kz_{2}+z_{1}}{m+n})$

What happens with if point M divides a line segment P ( x1, y1, z1) and Q ( x2, y2, z2 ) is the midpoint?

In this case, then m : n are in the ratio 1 : 1. That is the value of m = 1 and the value of n = 1.

$(\frac{1*x_{2}+1*x_{1}}{1+1}$, $\frac{1*y_{2}+1*y_{1}}{1+1}$, $\frac{1*z_{2}+1*z_{1}}{1+1})$

Therefore the coordinate of the midpoints are:

$(\frac{x_{2}+x_{1}}{m+n}$, $\frac{y_{2}+y_{1}}{m+n}$, $\frac{z_{2}+z_{1}}{m+n})$

### Solved Problems

Using the section formula in 3 dimension geometry, find the coordinates of the points J and K which divides the line segment PQ internally and externally both at the ratio 4 : 6? The coordinates of J are ( 3, 5, 2 ) and Q is (2, 4, 3).

Solution: The coordinates of  (dividing PQ internally), x = ( 4 × 2 + 6 × 3) ⁄ ( 4 + 6 ) = ( 8 + 18 )  ⁄ 10 = 26  ⁄  10, y = ( 4 × 4 + 6 × 5)  ⁄  ( 4 + 6 ) = ( 16 + 30 )  ⁄  10 = 46 ⁄ 8, z = ( 4 × 3 + 6 × 2)  ⁄  ( 4 + 6 ) = ( 12 + 12 ) ⁄ 10 = 24 ⁄ 10. The coordinates are ( 26 ⁄ 10, 46 ⁄ 10, 24 ⁄ 10 ).

Coordinates of L (dividing AB externally), The coordinates of  (dividing PQ internally), x = ( 4 × 2 - 6 × 3) ⁄ ( 4 - 6 ) = ( 8 - 18 )  ⁄ 10 = - 10  ⁄  10, y = ( 4 × 4 - 6 × 5)  ⁄  ( 4 - 6 ) = ( 16 - 30 )  ⁄  10 = -14  ⁄ 10, z = ( 4 × 3 - 6 × 2)  ⁄  ( 4 - 6 ) = ( 12 - 12 ) ⁄ 10 = 24 ⁄ 10. The coordinates are ( -10 ⁄ 10, -14 ⁄ 10, 0 ).

1) What are the Steps Involved in Section Formula? Prove it.

• Draw PU, MV, and QW perpendicular to XY plane such that PU || MV || QW as shown above.

• The points U, V and W lie on the straight line formed due to the intersection of a plane containing PU, MV and QW and XY- plane.

• From point M, a line segment KL is drawn such that it is parallel to UW.

• KL intersects PU externally at K and it intersects QW at L internally.

Since KL is parallel to UW and PU || MV || QW, therefore, the quadrilaterals UVMP and VWLM qualify as parallelograms.

Also, ∆PKM ~∆QLM therefore,

m/n = PM/QM = PK/QL = (KU-PU)/(QW-LW) = (VM-PU)/(QW-MV)

m/n = (z - z₁)/(z₂ - z)

z = (mz₂ + nz₁)/(m+n)

By drawing a perpendicular to the XZ plane and the YZ plane to find the coordinate x and coordinate y of point M divides the line PQ internally int the ratio m : n internally.,  you can repeat the above-stated procedure.

x = (mx₂ + nx₁)/(m+n) and y  = (yz₂ + yz₁)/(m+n).