. . Properties of the scalar triple product Theorem . For any three vectors , , a b and , ( .
Fig. . | cos r c r a r c Vector - - Applications of Vector Algebra Proof Let a = a i a j a k b i b j b k ˆ c i c j c k Then, = , by ↔ by ↔ = ( . Hence the theorem is proved.
Note By Theorem . , it follows that, in a scalar triple product, dot and cross can be interchanged without altering the order of occurrences of the vectors , by placing the parentheses in such a way that dot lies outside the parentheses, and cross lies between the vectors inside the parentheses. For instance, we have = ) , since dot and cross can be interchanged. = ( , since dot product is commutative.
) , since dot and cross can be interchanged = ( , since dot product is commutative = ) , since dot and cross can be interchanged Notation For any three vectors a b and c , the scalar triple product ( is denoted by [ , , ] a b c . [ , , ] a b c is read as box , , a b c . For this reason and also because the absolute value of a scalar triple product represents the volume of a box (rectangular parallelepiped),a scalar triple product is also called a box product . Note ( ) [ , , ] a b c = ( [ , , ] b c a [ , , ] b c a = ( [ , , ].
c a b In other words, [ , , ] [ , , ] [ , , ] a b c b c a c a b ; that is, if the three vectors are permuted in the same cyclic order, the value of the scalar