Vector Triple Product Definition:
Consider \overrightarrow a,\overrightarrow b,\overrightarrow c be the three vectors, then the vector triple product of vectors \overrightarrow a,\overrightarrow b,\overrightarrow c is defined as \overrightarrow a\times\left(\overrightarrow b\times\overrightarrow c\right)
Important Note:
Through given any three vectors \overrightarrow a,\overrightarrow b,\overrightarrow c 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 \overrightarrow a\times\left(\overrightarrow b\times\overrightarrow c\right)\neq\left(\overrightarrow a\times\overrightarrow b\right)\times\overrightarrow c for any vectors \overrightarrow a,\overrightarrow b,\overrightarrow c
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 \overrightarrow a,\overrightarrow b,\overrightarrow c we have \overrightarrow a\times\left(\overrightarrow b\times\overrightarrow c\right)\neq\left(\overrightarrow a.c\right)b\times\overrightarrow c+\left(\overrightarrow a.\overrightarrow b\right)\overrightarrow c
Proof :
Consider we choose the coordinate axes as follows :
Let x -axis is chosen along the line of action of \overrightarrow a vector, y -axis be chosen in the plane passing through vector \overrightarrow a and parallel to vector \overrightarrow b , and z-axis be chosen perpendicular to the plane containing vector \overrightarrow a and vector \overrightarrow b . Then, we have
Important Note :
(3) In \left( \overrightarrow a\times\overrightarrow b\right)\times\overrightarrow c , suppose the vectors inside the brackets, be called vector \overrightarrow b as the middle vector and vector \overrightarrow a as the non-middle vector. Similarly, in \overrightarrow a\times\left(\overrightarrow b\times\overrightarrow c\right) , vector \overrightarrow b is the middle vector and vector \overrightarrow c 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.
