. Scalar triple product Definition . For a given set of three vectors , a b , and c , the scalar ( is called a scalar triple product of , , a b c . Remark a b is a scalar and so ( a b has no meaning.
Vector - - Note Given any three vectors , a b and c , the following are scalar triple products: , ( , ( ), ), ), b a , ( , ( ), ), b a Geometrical interpretation of scalar triple product Geometrically, the absolute value of the scalar triple product ( is the volume of the parallelepiped formed by using the three vectors a b , , and c as co-terminus edges. Indeed, the magnitude of the vector ( is the area of the parallelogram formed by using a and b ; and the direction of the vector ( is perpendicular to the plane parallel to both a and b Therefore, | ( is | | | | | cos where θ is the angle between a and c .From Fig. . , we observe that | | | cos is the height of the parallelepiped formed by using the three vectors as adjacent vectors.
Thus, | ( is the volume of the parallelepiped. The following theorem is useful for computing scalar triple products. Theorem . If a = a i a j a k b b i b j b k ˆ c i c j c k , then = Proof By definition, we have = ⋅ a b a b i a b a b j a b a b k c i c j c k a b a b c a b a b c a b a b c which completes the proof of the theorem.