Question

Find the volume of the tetrahedron whose coterminous edges are
 \[A = \left( {1,2,3} \right)\] , \[B = \left( {3,7,4} \right)\] , \[C = \left( {5, - 2,3} \right)\] and \[D = \left( { - 4,5,6} \right)\] .

Answer 1

Hint: In this question, we have been given the coterminous edges of a tetrahedron and we have to calculate the volume of it. When there are 3 vectors, we calculate it directly by making the matrix of the three vectors and dividing the product by 6. But here, we have 4 vectors, so we solve this by finding the relative position of any three vectors with respect to the fourth vector. Then, we follow the same procedure as of the three vectors and find the volume of the tetrahedron.

Complete step-by-step answer:
We are given a tetrahedron with the following 4 coterminous edges:
$A = \left( {1,2,3} \right)$, $B = \left( {3,7,4} \right)$, $C = \left( {5, - 2,3} \right)$ and $D = \left( { - 4,5,6} \right)$
In this question, we are going to take $A$ as the base vector and find the relative positions of the other ones w.r.t. $A$.
\[
  \overrightarrow {AD} = D - A = \left( { - 4,5,6} \right) - \left( {1,2,3} \right) = \left( { - 5,3,3} \right) \\
  \overrightarrow {AC} = C - A = \left( {5, - 2,3} \right) - \left( {1,2,3} \right) = \left( {4, - 4,0} \right) \\
  \overrightarrow {AB} = B - A = \left( {3,7,4} \right) - \left( {1,2,3} \right) = \left( {2,5,1} \right) \;
 \]
Now, volume, \[V = \dfrac{1}{6}|\overrightarrow {AB} {\text{ }}\overrightarrow {AC} {\text{ }}\overrightarrow {AD} |\]
$
  V = \dfrac{1}{6}\left| {\begin{array}{*{20}{c}}
  2&5&1 \\
  4&{ - 4}&0 \\
  { - 5}&3&3
\end{array}} \right| \\
   = \dfrac{1}{6}\left( {2\left( {\left( { - 4 \times 3} \right) - 0} \right) - 5\left( {3 \times 4 - 0} \right) + 1\left( {4 \times 3 - 4 \times 5} \right)} \right) \\
   = \left| {\dfrac{1}{6}\left( { - 24 - 60 - 8} \right)} \right| = \dfrac{{92}}{6} = \dfrac{{46}}{3}
   = 15\dfrac{1}{3} \;
 $
Hence, the volume of the given tetrahedron is $15\dfrac{1}{3}$ cubic units.
So, the correct answer is “ \[15\dfrac{1}{3}\] cubic units”.

Note: So, we saw that in solving questions like these, we first need to see how many variables or how many vectors are there in the question. If there are 3 vectors, then we can calculate it directly by forming a matrix of the three vectors. And when we have the product after the matrix multiplication, we divide the product by 6. Then, if there are 4 vectors, we solve that representation of the tetrahedron by taking any one vector as the base vector and we find the relative position of the remaining three vectors with respect to the first, base vector. And then we follow the same procedure as we do in the three vectors’ case and we find the volume of the given tetrahedron.