Final answer:
The volume of the parallelepiped determined by vectors a, b, and c is found by calculating the scalar triple product, which is the dot product of vector a with the cross product of vectors b and c. The result is 10 cubic units.
Step-by-step explanation:
The volume of the parallelepiped determined by the vectors a = <1, -2, 3>, b = <2, 4, 2>, and c = <2, 1, 4> can be found by calculating the scalar triple product of these vectors. This product is given by the dot product of one of the vectors with the cross product of the other two. In this case, we can find the volume by calculating (b x c) · a.
Step-by-step calculation:
- First, calculate the cross product of vectors b and c: b x c.
- Then, take the dot product of vector a with the result from step 1: (b x c) · a.
- The absolute value of the result from step 2 is the volume of the parallelepiped.
Let's perform the calculations:
- b x c = - +
= <16i - 6j -6k> - (b x c) · a = <16, -6, -6> · <1, -2, 3>
= 16*1 - 6*(-2) + (-6)*3
= 16 + 12 - 18
= 10
The absolute value of 10 is 10, so the volume of the parallelepiped is 10 cubic units.