Find the volume of the tetrahedron having the given vertices. (3, - 1, 1), (5, - 2, 4), (1, 1, 1), (0, 0, 1)

Given four vertices of a tetrahedron, we need to find its volume.

A (3,-1,1), B (5,-2,4), C (1,1,1), D (0,0,1)

The simplest method is to use vectors.

Here, we calculate the volume of a parallelepiped defined by vectors AB, AC, AD. The volume of a tetrahedron is one-sixth of the volume of the parallelepiped, V, given by AB. (ACxAD).

Here are the steps:

1. Choose a vertex, say A. Calculate vectors AB, AC and AD

AB = (5-2,4) - (3,-1,1) =

AC = (1,1,1) - (3,-1,1) =

AD = (0,0,1) - (3,-1,1) =

2. Calculate vector cross product ACxAD = x

i j k

-2 - 2 0

-3, 1, 0

=

=

3. Calculate volume of parallelepiped, equal to the dot product AB. (ACxAD), ignore sign, which is arbitrary.

V = |.| = |12| = 12

4. Calculate volume of tetrahedron, which is one-sixth of volume of parallelepiped

V (tetrahedron) = V/6 = 12/6 = 2