Final answer:
To find the volume of a parallelepiped, we can use the scalar triple product formula. By finding the cross product of vectors b and c, and then taking the dot product with vector a, we can calculate the volume of the parallelepiped. In this scenario, the volume is 8.
Step-by-step explanation:
To find the volume of a parallelepiped determined by vectors a, b, and c, you can use the scalar triple product. The scalar triple product is defined as (B x C) · A, where B and C are the cross products of vectors b and c, and a is the dot product of vector A with the cross product of vectors b and c. In this case, a = (1, 2, 4), b = (-1, 1, 2), and c = (5, 1, 3). So, we need to find the cross product of b and c, then take the dot product with a. The scalar triple product gives us the volume of the parallelepiped.
Cross product: B = b x c = (-1, 1, 2) x (5, 1, 3) = (-5, -1, 6) - (2, -7, 6) = (-7, -6, 12)
Dot product: (B x C) · A = (-7, -6, 12) · (1, 2, 4) = -7(1) + -6(2) + 12(4) = 8.