LinAlg4.2.21

From Exampleproblems

Find the volume of the tetrahedron with vertices $(1,1,3),(4,3,2),(5,2,7),(6,4,8)\,$

Let $A=\bar{i}+\bar{j}+3\bar{k},B=4\bar{i}+3\bar{j}+2\bar{k},C=5\bar{i}+2\bar{j}+7\bar{k},D=6\bar{i}+4\bar{j}+8\bar{k}\,$

Now $\bar{AB}=3\bar{i}+2\bar{j}+\bar{k}.\bar{AC}=4\bar{i}+\bar{j}+4\bar{k},\bar{AD}=5\bar{i}+3\bar{j}+5\bar{k}\,$

Therefore,volume of tetrahedron is $\frac{1}{6}|[\bar{AB} \bar{AC} \bar{AD}]|=\frac{1}{6}\begin{vmatrix} 3 & 2 & 1 \\ 4 & 1 & 4 \\ 5 & 3 & 5 \end{vmatrix} \,$ which is equal to $\frac{1}{6}|3(5-12)-2(20-20)+1(12-5)|=\frac{1}{6}|-21+7|=\frac{14}{6}=\frac{7}{3}\,$ cubic units.