Final answer:
The volume of the parallelepiped formed by vectors a, b, and c is found using the scalar triple product, which involves calculating cross product b x c and taking the dot product with a. The result is the absolute value of the scalar triple product, which is |-32|, hence the volume is 32 cubic units.
Step-by-step explanation:
To find the volume of the parallelepiped determined by the vectors a, b, and c, we can use the scalar triple product, which is the dot product of one of the vectors with the cross product of the other two. For vectors a = (1, 3, 4), b = (-1, 1, 3), and c = (3, 1, 2), we first calculate the cross product b x c, and then take the dot product of that result with vector a.
The cross product b x c is:
- i(1*2 - 3*1) - j(1*2 - (-1)*3) + k(-1*1 - 1*3)
- i(2 - 3) - j(2 + 3) + k(-1 - 3)
- -i - 5j - 4k
The dot product (b x c) . a is:
= (-i - 5j - 4k) . (i + 3j + 4k)
= -(1*1) - (5*3) - (4*4)
= -1 - 15 - 16
= -32
The absolute value of the scalar triple product gives us the volume of the parallelepiped. Therefore, the volume is |-32| = 32 cubic units.