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Harlamova29_29 [7]
3 years ago
11

The volume of a tetrahedron is 1 6 of the volume of a parallelepiped whose sides are formed using the vectors coming out of one

corner of the tetrahedron. Find the volume of the tetrahedron with corners at (1, 1, 1), (1, 5, 5), (2, 1, 3), and (2, 2, 1).
Mathematics
1 answer:
Irina18 [472]3 years ago
5 0

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

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