Final answer:
The scalar triple product (B × C) · A is equal to the volume of the parallelepiped formed by vectors A, B, and C, calculated by the dot product of the cross product B × C (area of base) with A (height).
Step-by-step explanation:
The student has asked to show that the scalar triple product (B × C) · A, also known as the dot product of a cross product, represents the volume of a parallelepiped formed by vectors A, B, and C. The scalar triple product can be used to determine the volume of a parallelepiped whose adjacent edges are given by three vectors.
According to the vector calculus, the cross product B × C gives a vector that is perpendicular to the plane containing B and C, and its magnitude is equal to the area of the parallelogram spanned by these vectors. When you take the dot product of this result with vector A, you are effectively projecting A onto the direction of B × C (which is the normal of the base of the parallelepiped) and multiplying the projected length of A (which represents the height of the parallelepiped) with the area of the base. Mathematically, this projection is precisely the height h that, when multiplied by the area of the base A, gives the volume V of the parallelepiped (V = A ⋅ h).
To prove this concept, you can imagine the vectors A, B, and C as three edges of a parallelepiped that meet at one corner. The bottom face of the parallelepiped is the parallelogram created by B and C, and A points diagonally up from this corner. The magnitude of B × C represents the area of the parallelogram (the base), and when you multiply it by the scalar component of A in the direction normal to the base (the dot product (B × C) · A), you get the total volume. Therefore, ((B × C) · A) equals the volume of the parallelepiped.