Calculate the volume of a tetrahedron with vertices at A(1, 2, 3), B(4, 5, 6), C(7, 8, 9), and D(10, 11, 12).
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To calculate the volume of a tetrahedron, we can use the formula:
Volume = (1/6) * |(AB · AC) × AD|
where AB represents the vector from A to B, AC represents the vector from A to C, AD represents the vector from A to D, and × denotes the cross product.
First, let's calculate the vectors AB, AC, and AD:
AB = B - A = (4, 5, 6) - (1, 2, 3) = (3, 3, 3)
AC = C - A = (7, 8, 9) - (1, 2, 3) = (6, 6, 6)
AD = D - A = (10, 11, 12) - (1, 2, 3) = (9, 9, 9)
Next, let's calculate the dot product AB · AC:
AB · AC = (3, 3, 3) · (6, 6, 6) = 3*6 + 3*6 + 3*6 = 54
Now, let's calculate the cross product (AB · AC) × AD:
(AB · AC) × AD = (54, 54, 54) × (9, 9, 9)
To calculate the cross product, we can use the formula:
(a, b, c) × (d, e, f) = (bf - ce, cd - af, ae - bd)
Using this formula, we get:
(AB · AC) × AD = ((54*9) - (54*9), (54*9) - (54*9), (54*9) - (54*9)) = (0, 0, 0)
Since the cross product is zero, the volume of the tetrahedron is also zero.