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.
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 = \[\frac{my_{2}+ny_{1}}{m+n}\]
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:
\[(\frac{mx_{2}+nx_{1}}{m+n}\], \[\frac{my_{2}+ny_{1}}{m+n}\], \[\frac{mz_{2}+nz_{1}}{m+n})\]
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:
\[(\frac{x_{2}+x_{1}}{m+n}\], \[\frac{y_{2}+y_{1}}{m+n}\], \[\frac{z_{2}+z_{1}}{m+n})\]
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).
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