Geometrical Interpretation - Scalar Triple Product Proof

Suppose there exists a parallelepiped with vectors a, b and c along sides OA, OB and OC respectively.

Height = OA

= a cos

where angle which the height OA makes with the base of the parallelepiped is

Area of base = area of parallelogram OBDC

= | b * c | (from definition of cross product)

= |b| |c| sin
where angle between OB and OC is
Volume of parallelepiped = Area of base * Height
= Area of parallelogram OBDC * OA
= (|b| |c| sin theta) * ( |a| cos alpha)
= ( |b| * |c| ) a cos= a . ( b * c)