Answer: The volume of the tetrahedron is 2 units.
Step-by-step explanation:
Let A = (1, 1, 1)
B = (1, 5, 5)
C = (2, 2, 1)
The volume of a tetrahedron is given as
V = (1/6)|AB, AC, AD|
Where |AB, AC, AD| is the determinant of the matrix of AB, AC, AD.
We need to determine AB, AC, and AD
Suppose A = (a1, a2, a3)
B = (b1, b2, b3)
C = ( c1, c2, c3)
AB = ( b1 - a1, b2 - a2, b3 - a3)
Similarly for AB, AC, BC, etc.
AB = (1 - 1, 5 - 1, 5 - 1)
= (0, 4, 4)
AC = (1, 0, 2)
AD = (1, 1, 0)
Volume =
(1/6) |0 4 4|
|1 0 2|
|1 1 0|
= (1/6)[0(0 - 2) - 4(0 - 2) + 4(1 - 0)
= (1/6)(0 + 8 - 4)
= (1/6)(12)
V = 12/6 = 2 units
