Vector Triple Product Definition:
Consider 
be the three vectors, then the vector triple product of vectors is defined as 
Important Note:
Through given any three vectors the following products are the vector triple products :
Employing the well-known properties of the vector product, we get the following theorems.
Theorem 1 :
It satisfies the properties which are given below.
Result:
The VTP does not obey associative law. This means that 
for any vectors
Reason:
The following theorem will give a simple formula to evaluate the vector triple product.
Theorem 2: (Vector Triple Product Expansion)
For any three vectors we have 
Proof :
Consider we choose the coordinate axes as follows :
Let 
-axis is chosen along the line of action of 
vector, 
-axis be chosen in the plane passing through vector and parallel to vector 
, and 
-axis be chosen perpendicular to the plane containing vector and vector . Then, we have
Important Note :
(3) In 
, suppose the vectors inside the brackets, be called vector as the middle vector and vector as the non-middle vector. Similarly, in , vector is the middle vector and vector 
is the non-middle vector. Then we analyze 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.