Final answer:
The subject is Mathematics, focusing on geometry and vector analysis. The scalar triple product (B x C) · A is used to find the volume of the parallelepiped formed by vectors A, B, and C.
Step-by-step explanation:
The question involves a geometrical concept where point P is inside parallelogram ABCD. The student is required to show that the volume of a parallelepiped formed by three vectors can be calculated as the scalar triple product of those vectors, which is notated as (B x C) · A. Here A, B, and C represent the vectors corresponding to edges of the parallelepiped. To determine the volume of a parallelepiped using vector analysis one might use the scalar triple product, which involves a combination of vector cross product and dot product. The scalar triple product gives a scalar value which equals the volume of the parallelepiped defined by the three vectors.
In terms of areas, the question also includes the concepts of areas within triangles A₁ for triangle ABP and A₂ for triangle CDP. However, to resolve the volume issue as initially asked, one must utilize vector operations rather than just area calculations.