Question

Find the volume of the tetrahedron having the vertices (4.-1, 1) (9.-9, 9) (1, 1, 1) and (0,0,3)

Answer 1

The volume of the tetrahedron with those vertices is 2.

Explanation:

The volume of the tetrahedron can be found by the mixed product of three vectors u,v,w divided by 6. So

`V_(T) = \frac{|u.v \text{x} w|}{6}`

`|u.v \text{x} w|` 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)

`|u.v \text{x} w| = det \left[\begin{array}{ccc}5&-8&8\\-3&2&0\\-4&1&2\end{array}\right] = 12`

`V_(T) = \frac{|u.v \text{x} w| }{6} = (12)/(6) = 2`

The volume of the tetrahedron with those vertices is 2.