Question

Find the volumn of tetrahedron whose vertices are $A(1,1)B(-4,3,6)C(-1,0,3)$ & $D(2,4,-5)$

Answer 1

The volume of a tetrahedron with vertices a,b,c and d is given by

$V=\frac{|(a-d)\cdot \left(\right(b-c)\times (c-d)|}{6}$

Thus, for $A(1,1,0),B(-4,3,6),C(-1,0,3)andD(2,4,-5)$

$V=\frac{|(-1,-3,5)\cdot \left(\right(-3,3,3)\times (-3,-4,8)|}{6}$

$(-3,3,3)\times (-3,-4,8)=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -3& 3& 3\\ -3& -4& 8\end{array}\right|$

$=\hat{i}\times 3\times 8+\hat{j}\times 3\times (-3)+\hat{k}\times (-3)\times (-4)-(-3)\times 3\times \hat{k}-(-4)\times 3\times \hat{i}-8\times (-3)\times \hat{j}$

$=36\hat{i}+15\hat{j}+21\hat{k}=(36,15,21)$

$V=\frac{|(-1,-3,5)\cdot (36,15,21)|}{6}$

$V=\frac{|(-36-30+105)|}{6}$

$V=\frac{|39|}{6}$

$V=6.5$