Question

Volume of the tetrahedron whose vertices are represented by the position vectors, \[A(0,1,2);\,B(3,0,1);\,C(4,3,6)\] and \[D(2,3,2)\] is
A. \[3\]
B. \[6\]
C. \[36\]
D.None

Answer 1

Hint: We are asked to find the volume of the tetrahedron. For this, recall the formula for volume of a tetrahedron. The vertices of the tetrahedron are given, draw a tetrahedron showing the vertices and find the edges. Using these edges find the volume of the tetrahedron

Complete step-by-step answer:
Given, the vertices of the tetrahedron are \[A(0,1,2);\,B(3,0,1);\,C(4,3,6)\] and \[D(2,3,2)\]
Let us draw a tetrahedron with the given vertices,

In the form of vectors, the points can be written as,
 \[\overrightarrow {\text{A}} = 0\widehat {\text{i}} + 1\widehat {\text{j}} + 2\widehat {\text{k}}\]
 \[\overrightarrow {\text{B}} = 3\widehat {\text{i}} + 0\widehat {\text{j}} + 1\widehat {\text{k}}\]
 \[\overrightarrow {\text{C}} = 4\widehat {\text{i}} + 3\widehat {\text{j}} + 6\widehat {\text{k}}\]
 \[\overrightarrow {\text{D}} = 2\widehat {\text{i}} + 3\widehat {\text{j}} + 2\widehat {\text{k}}\]
Volume of a tetrahedron can be expressed as,
 \[V = \dfrac{1}{6}\left| {\overrightarrow a \cdot (\overrightarrow b \times \overrightarrow c )} \right|\] (i)
where \[a\] , \[b\] and \[c\] are three non coplanar vectors which represent the four coterminous edges.

Now, let us find the edges of the given tetrahedron
 \[\overrightarrow {{\text{AB}}} = \overrightarrow {\text{B}} - \overrightarrow {\text{A}} = (3 - 0)\widehat {\text{i}} + (0 - 1)\widehat {\text{j}} + (1 - 2)\widehat {\text{k}} = 3\widehat {\text{i}} - 1\widehat {\text{j}} - 1\widehat {\text{k}}\]
 \[\overrightarrow {{\text{AC}}} = \overrightarrow {\text{C}} - \overrightarrow {\text{A}} = (4 - 0)\widehat {\text{i}} + (3 - 1)\widehat {\text{j}} + (6 - 2)\widehat {\text{k}} = 4\widehat {\text{i}} + 2\widehat {\text{j}} + 4\widehat {\text{k}}\]
 \[\overrightarrow {{\text{AD}}} = \overrightarrow {\text{D}} - \overrightarrow {\text{A}} = (2 - 0)\widehat {\text{i}} + (3 - 1)\widehat {\text{j}} + (2 - 2)\widehat {\text{k}} = 2\widehat {\text{i}} + 2\widehat {\text{j}} + 0\widehat {\text{k}} = 2\widehat {\text{i}} + 2\widehat {\text{j}}\]
Vectors \[\overrightarrow {{\text{AB}}} \] , \[\overrightarrow {{\text{AC}}} \] and \[\overrightarrow {{\text{AD}}} \] are three non coplanar vectors. Therefore, volume of the tetrahedron using the formula from equation (i) is,
 \[V = \dfrac{1}{6}\left| {\overrightarrow {{\text{AB}}} \cdot (\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AD}}} )} \right|\] (ii)
Let us solve this part by part.
First let us take \[\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AD}}} \] ,
Putting the values of \[\overrightarrow {{\text{AC}}} \] and \[\overrightarrow {{\text{AD}}} \] we get,
 \[\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AD}}} = \left[ {\left( {4\widehat {\text{i}} + 2\widehat {\text{j}} + 4\widehat {\text{k}}} \right) \times \left( {2\widehat {\text{i}} + 2\widehat {\text{j}}} \right)} \right] \\
   = \left[ {\begin{array}{*{20}{c}}
  {\widehat {\text{i}}}&{\widehat {\text{j}}}&{\widehat {\text{k}}} \\
  4&2&4 \\
  2&2&0
\end{array}} \right] \]
 \[= (2 \times 0 - 4 \times 2)\widehat {\text{i}} - (4 \times 0 - 4 \times 2)\widehat {\text{j}} + (4 \times 2 - 2 \times 2)\widehat {\text{k}} \\
   = - 8\widehat {\text{i}} + 8\widehat {\text{j}} + 4\widehat {\text{k}} \]
Now, putting the value of \[\overrightarrow {{\text{AB}}} \] and \[\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AD}}} \] in equation (ii), we get
 \[V = \dfrac{1}{6}\left| {\left( {3\widehat {\text{i}} - 1\widehat {\text{j}} - 1\widehat {\text{k}}} \right) \cdot \left( { - 8\widehat {\text{i}} + 8\widehat {\text{j}} + 4\widehat {\text{k}}} \right)} \right| \\
   \Rightarrow V = \dfrac{1}{6}\left| { - 24 - 8 - 4} \right| \\
   \Rightarrow V = \dfrac{1}{6}\left| { - 36} \right| \\
   \Rightarrow V = \dfrac{1}{6} \times 36 = 6\,{\text{unit}}\,{\text{cube}} \]
Therefore, the volume of the tetrahedron is \[6\,{\text{unit}}\,{\text{cube}}\] .
So, the correct answer is “Option C”.

Note: There are few more important geometrical shapes in three-dimension, these are cube, cuboid, pentagonal prism, hexagonal prism, octahedron, cone, cylinder, ellipsoid. The number of vertices, faces and edges are different for different shapes. Students usually get confused between the number of vertices and edges of different shapes which leads to an incorrect answer so, always try to remember the number of vertices, edges and faces for each shape.