JEE Main & Advanced Mathematics Three Dimensional Geometry Section Formula

(1) Section formula for internal or external division : Let \[P({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[Q({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] be two points. Let \[R\] be a point on the line segment joining \[P\] and \[Q\] such that it divides the join of \[P\] and \[Q\] internally or externally in the ratio \[{{m}_{1}}:{{m}_{2}}\].

Then the co-ordinates of \[R\] are

\[\left( \frac{{{m}_{1}}{{x}_{2}}\pm {{m}_{2}}{{x}_{1}}}{{{m}_{1}}\pm {{m}_{2}}},\,\frac{{{m}_{1}}{{y}_{2}}\pm {{m}_{2}}{{y}_{1}}}{{{m}_{1}}\pm {{m}_{2}}},\,\frac{{{m}_{1}}{{z}_{2}}\pm {{m}_{2}}{{z}_{1}}}{{{m}_{1}}\pm {{m}_{2}}} \right)\].

(2) Co-ordinates of the general point : The co-ordinates of any point lying on the line joining points \[P({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[Q({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] may be taken as \[\left( \frac{k{{x}_{2}}+{{x}_{1}}}{k+1},\,\frac{k{{y}_{2}}+{{y}_{1}}}{k+1},\,\frac{k{{z}_{2}}+{{z}_{1}}}{k+1} \right)\], which divides \[PQ\] in the ratio \[k:1\]. This is called general point on the line \[PQ\].