. Vector triple product Definition . For a given set of three vectors , , a b c , the vector ) is called a vector triple product . Note Given any three vectors a b c , , the following are vector triple products : , ( , ( ), ), b c × (( Using the well known properties of the vector product, we get the following theorem.
Theorem . The vector triple product satisfies the following properties. ( ) ) ( = ∈ ), ( ) ( )), ( ) (( = ∈ ), (( )), ( ) )) ∈ ), )) )), Remark Vector triple product is not associative . This means that ≠ , for some vectors , , a b c .
Justification We take i b i c . Then, = × = × but ˆ Therefore, ≠ . The following theorem gives a simple formula to evaluate the vector triple product. Theorem .
(Vector Triple product expansion) For any three vectors , , a b c we have a c b a b c . Proof Let us choose the coordinate axes as follows : Let x -axis be chosen along the line of action of , a y -axis be chosen in the plane passing through a and parallel to b , and z -axis be chosen perpendicular to the plane containing a and b . Then, we have Vector - - Applications of Vector Algebra a = ˆ a i b i b j c = ˆ c i c j c k From equations ( ) and ( ), we get = ( a c b a b c Note ( ) β , where α = a c and β = − a b , and so it lies in the plane parallel to b and c . ( ) We also note that × = −× = − {( ) } c b a c a b = ( a c b b c a Therefore, ( × lies in the plane parallel to a and b .
( ) In ( × , consider the vectors inside the brackets, call b as the middle vector and a as the non-middle vector. Similarly, in ), is the middle vector and c is the non-middle vector. Then we observe that a vector triple product of these vectors is equal to λ (middle vector) −µ (non-middle vector) where λ is the dot product of the vectors other than the middle vector and μ is the dot product of the vectors other than the non-middle vector.