LinAlg4.2.21

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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.


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