Respuesta :
Answer:
The volume of the tetrahedron with those vertices is 2.
Step-by-step explanation:
The volume of the tetrahedron can be found by the mixed product of three vectors u,v,w divided by 6. So
[tex]V_{T} = \frac{|u.v \text{x} w|}{6}[/tex]
[tex]|u.v \text{x} w|[/tex] is the determinant of a matrix in which each line is formed by the elements of those vectors.
With four points, we can have find three vectors.
We have those following points:
A(4,-1, 1)
B(9.-9, 9)
C(1, 1, 1)
D(0,0,3)
I am going to form those following vectors:
u = AB = B - A = (9,-9,9)-(4,-1,1) = (5, -8, 8)
v = AC = C - A = (1,1,1)-(4,-1,1) = (-3, 2, 0)
w = AD = D - A = (0,0,3) - (4,-1,1) = (-4,1,2)
[tex]|u.v \text{x} w| = det \left[\begin{array}{ccc}5&-8&8\\-3&2&0\\-4&1&2\end{array}\right] = 12[/tex]
[tex]V_{T} = \frac{|u.v \text{x} w| }{6} = \frac{12}{6} = 2[/tex]
The volume of the tetrahedron with those vertices is 2.
