JEE Main & Advanced Mathematics Three Dimensional Geometry Triangle and Tetrahedron

(1) Co-ordinates of the centroid

(i) If \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\,({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] and \[({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})\] are the vertices of a triangle, then co-ordinates of its centroid are \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\,\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3},\,\frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}{3} \right)\]

(ii) If \[({{x}_{r}},\,{{y}_{r}},\,{{z}_{r}})\]; \[r=\text{ }1,\text{ }2,\text{ }3,\text{ }4,\] are vertices of a tetrahedron, then co-ordinates of its centroid are \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}}{4},\,\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}}{4},\,\frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}+{{z}_{4}}}{4} \right)\]

(2) Area of triangle : Let \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\], \[B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] and \[C({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})\] be the vertices of a triangle, then \[{{\Delta }_{x}}=\frac{1}{2}\left| \,\begin{matrix} {{y}_{1}} & {{z}_{1}} & 1  \\ {{y}_{2}} & {{z}_{2}} & 1  \\ {{y}_{3}} & {{z}_{3}} & 1  \\ \end{matrix}\, \right|\], \[{{\Delta }_{y}}=\frac{1}{2}\left| \,\begin{matrix} {{x}_{1}} & {{z}_{1}} & 1  \\ {{x}_{2}} & {{z}_{2}} & 1  \\ {{x}_{3}} & {{z}_{3}} & 1  \\ \end{matrix}\, \right|\], \[{{\Delta }_{z}}=\frac{1}{2}\left| \,\begin{matrix} {{x}_{1}} & {{y}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & 1  \\ \end{matrix}\, \right|\]

Now, area of \[\Delta ABC\] is given by the relation \[\Delta =\sqrt{\Delta _{x}^{2}+\Delta _{y}^{2}+\Delta _{z}^{2}}\].

(3) Condition of collinearity: Points \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\] \[B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] and \[C({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})\] are collinear,

If  \[\frac{{{x}_{1}}-{{x}_{2}}}{{{x}_{2}}-{{x}_{3}}}=\frac{{{y}_{1}}-{{y}_{2}}}{{{y}_{2}}-{{y}_{3}}}=\frac{{{z}_{1}}-{{z}_{2}}}{{{z}_{2}}-{{z}_{3}}}\].

(4) Volume of tetrahedron : If vertices of tetrahedron be  \[({{x}_{r}},\,{{y}_{r}},\,{{z}_{r}})\]; \[r=\text{ }1,\text{ }2,\text{ }3,\text{ }4;\] then \[V=\frac{1}{6}\left| \,\begin{matrix} {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1  \\ {{x}_{4}} & {{y}_{4}} & {{z}_{4}} & 1  \\ \end{matrix}\, \right|\].