Final answer:
The volume of a parallelepiped can be calculated using the scalar triple product, which is a geometric representation of volume calculated from the cross product of two vectors dot multiplied with the third vector.
Step-by-step explanation:
To find the volume of a parallelepiped defined by three vectors originating from a common vertex, you can use the scalar triple product. In this case, the vectors are A = (1,0,-3), B = (1,2,4), and C = (5,1,0), that serve as the edges of the parallelepiped. The scalar triple product is given by (B x C) · A, which geometrically represents the volume of the parallelepiped.
The cross product of vectors B and C, B x C, is calculated first, and then the result is dot multiplied (denoted by ·) with vector A. This process yields a scalar value which represents the required volume.
Using the formula:
- B x C = (B2C3 - B3C2, B3C1 - B1C3, B1C2 - B2C1)
- (B x C) · A = A1(B2C3 - B3C2) + A2(B3C1 - B1C3) + A3(B1C2 - B2C1)
Carrying out the calculations will give us the volume of the parallelepiped.