Final answer:
The volume of the parallelepiped defined by vectors a, b, and c is found using the scalar triple product. For vectors a = (1, 3, 2), b = (-1, 1, 2), and c = (3, 1, 3), the volume calculated is 20 cubic units.
Step-by-step explanation:
The volume of a parallelepiped can be calculated using the scalar triple product of its defining vectors. In this case, we want to find the volume of the parallelepiped determined by vectors a, b, and c. To do this, we calculate the product (b x c).a, where 'x' represents the cross product and the dot represents the scalar (dot) product.
First, we find the cross product of vectors b and c:
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- b = (-1, 1, 2)
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- c = (3, 1, 3)
Cross product b x c yields (1*3 - 2*1, -((-1)*3 - 2*3), (-1)*1 - 1*3) = (1, 9, -4).
Now, applying the dot product with vector a = (1, 3, 2) will give us the volume:
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- V = (1, 9, -4) · (1, 3, 2) = 1*1 + 9*3 + -4*2
The calculation results in V = 1 + 27 - 8 = 20 cubic units. Therefore, the volume of the parallelepiped is 20 cubic units.