[Maths Class Notes] on Section Formula in 3 Dimension Pdf for Exam

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.

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

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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 ).