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
