`\color{red} ✍️` In two dimensional geometry, we have learnt how to find the coordinates of a point dividing a line segment in a given ratio internally. Now, we extend this to three dimensional geometry as follows:
`\color{red} ✍️` Let the two given points be `P(x_1, y_1, z_1)` and `Q (x_2, y_2, z_2).`
`\color{red} ✍️` Let the point `R (x, y, z)` `PL, QM` and `RN` perpendicular to the `XY`-plane.
Obviously `PL ∥ RN ∥ QM` and feet of these perpendiculars lie in a `XY`-plane.
The points `L, M` and `N` will lie on a line which is the intersection of the plane containing `PL, RN` and `QM` with the `XY`-plane.
Through the point R draw a line `ST` parallel to the line `LM`. Line `ST` will intersect the line `LP` externally at the point `S` and the line `MQ` at `T`, as shown in Fig 12.5.
`\color{red} ✍️` Also note that quadrilaterals `LNRS` and `NMTR` are parallelograms.
`\color{red} ✍️` The triangles `PSR` and `QTR` are similar. Therefore,
`m/n = (PR)/(QR) = (SP)/(QT) = (SL - PL)/(QM - TM) = (NR - PL)/(QM - NR) = (z-z_1)/(z_2-z)`
This implies `color{fuchsia}(z = ( m z_2+ n z_1)/(m+n))`
`\color{red} ✍️` Similarly, by drawing perpendiculars to the XZ and YZ-planes, we get
`color{fuchsia}(y = ( my_2+n y_1)/(m+n))` and `color{fuchsia}(x = (m x_2+n x_1)/(m+n))`
Hence, the coordinates of the point `R` which divides the line segment joining two points `P (x_1, y_1, z_1)` and `Q (x_2, y_2, z_2)` internally in the ratio `m : n` are
`color{red}((( mx_2- nx_1)/(m-n) , (my_2-ny_1)/(m-n) , (mz_2-nz_1)/(m-n)))`
`color(blue)("Case 1 :")` Coordinates of the mid-point: In case `R` is `color(blue)("the mid-point of PQ,")` then
`color{fuchsia}(m : n = 1 : 1 )` so that `color{fuchsia}(x = (x_1+x_2)/2 , y = (y_1+y_2)/2)` and `color{fuchsia}(z = (z_1+z_2)/2)`
These are the coordinates of the mid point of the segment joining `P( x_1 , y_1 , z_1)` and `Q ( x_2 , y_2 , z_2)`
`color(blue)("Case 2 :")` The coordinates of the point `R` which divides `PQ` in the ratio `k : 1` are obtained by taking `k = m/n` which are as given below:
`color{fuchsia}((( kx_2+x_1)/(1+k) , (ky_2+y_1)/(1+k) , (kz_2+z_1)/(1+k)))`
Generally, this result is used in solving problems involving a general point on the line passing through two given points.
`\color{red} ✍️` In two dimensional geometry, we have learnt how to find the coordinates of a point dividing a line segment in a given ratio internally. Now, we extend this to three dimensional geometry as follows:
`\color{red} ✍️` Let the two given points be `P(x_1, y_1, z_1)` and `Q (x_2, y_2, z_2).`
`\color{red} ✍️` Let the point `R (x, y, z)` `PL, QM` and `RN` perpendicular to the `XY`-plane.
Obviously `PL ∥ RN ∥ QM` and feet of these perpendiculars lie in a `XY`-plane.
The points `L, M` and `N` will lie on a line which is the intersection of the plane containing `PL, RN` and `QM` with the `XY`-plane.
Through the point R draw a line `ST` parallel to the line `LM`. Line `ST` will intersect the line `LP` externally at the point `S` and the line `MQ` at `T`, as shown in Fig 12.5.
`\color{red} ✍️` Also note that quadrilaterals `LNRS` and `NMTR` are parallelograms.
`\color{red} ✍️` The triangles `PSR` and `QTR` are similar. Therefore,
`m/n = (PR)/(QR) = (SP)/(QT) = (SL - PL)/(QM - TM) = (NR - PL)/(QM - NR) = (z-z_1)/(z_2-z)`
This implies `color{fuchsia}(z = ( m z_2+ n z_1)/(m+n))`
`\color{red} ✍️` Similarly, by drawing perpendiculars to the XZ and YZ-planes, we get
`color{fuchsia}(y = ( my_2+n y_1)/(m+n))` and `color{fuchsia}(x = (m x_2+n x_1)/(m+n))`
Hence, the coordinates of the point `R` which divides the line segment joining two points `P (x_1, y_1, z_1)` and `Q (x_2, y_2, z_2)` internally in the ratio `m : n` are
`color{red}((( mx_2- nx_1)/(m-n) , (my_2-ny_1)/(m-n) , (mz_2-nz_1)/(m-n)))`
`color(blue)("Case 1 :")` Coordinates of the mid-point: In case `R` is `color(blue)("the mid-point of PQ,")` then
`color{fuchsia}(m : n = 1 : 1 )` so that `color{fuchsia}(x = (x_1+x_2)/2 , y = (y_1+y_2)/2)` and `color{fuchsia}(z = (z_1+z_2)/2)`
These are the coordinates of the mid point of the segment joining `P( x_1 , y_1 , z_1)` and `Q ( x_2 , y_2 , z_2)`
`color(blue)("Case 2 :")` The coordinates of the point `R` which divides `PQ` in the ratio `k : 1` are obtained by taking `k = m/n` which are as given below:
`color{fuchsia}((( kx_2+x_1)/(1+k) , (ky_2+y_1)/(1+k) , (kz_2+z_1)/(1+k)))`
Generally, this result is used in solving problems involving a general point on the line passing through two given points.