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) = <2, -1, 3>
AC=(1,1,1)-(3,-1,1) = <-2, 2, 0>
AD=(0,0,1)-(3,-1,1) = <-3, 1, 0>
2. Calculate vector cross product ACxAD = <-2,2,0>x<-3,1,0>
  i   j   k
-2 -2  0
-3, 1, 0
=<0-0, 0-0, -2-(-6)>
= <0, 0, 4>
3. Calculate volume of parallelepiped, equal to the dot product AB.(ACxAD), ignore sign, which is arbitrary.
V = |<2, -1, 3>.<0,0,4>| = |12| = 12
4. Calculate volume of tetrahedron, which is one-sixth of volume of parallelepiped
V(tetrahedron) = V/6= 12/6 = 2