Vector Triple Product Definition:
Consider [katex]\overrightarrow a,\overrightarrow b,\overrightarrow c[/katex] be the three vectors, then the vector triple product of vectors [katex]\overrightarrow a,\overrightarrow b,\overrightarrow c[/katex] is defined as [katex] \overrightarrow a\times\left(\overrightarrow b\times\overrightarrow c\right)[/katex]
Important Note:
Through given any three vectors [katex]\overrightarrow a,\overrightarrow b,\overrightarrow c[/katex] 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 [katex]\overrightarrow a\times\left(\overrightarrow b\times\overrightarrow c\right)\neq\left(\overrightarrow a\times\overrightarrow b\right)\times\overrightarrow c [/katex] for any vectors [katex]\overrightarrow a,\overrightarrow b,\overrightarrow c[/katex].
Theorem 2: (Vector Triple Product Expansion)
For any three vectors [katex]\overrightarrow a,\overrightarrow b,\overrightarrow c[/katex] we have [katex]\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 [/katex]
Proof :
Consider we choose the coordinate axes as follows :
Let [katex]x[/katex] -axis is chosen along the line of action of [katex] \overrightarrow a[/katex] vector, [katex]y[/katex] -axis be chosen in the plane passing through vector [katex] \overrightarrow a[/katex] and parallel to vector [katex] \overrightarrow b[/katex], and [katex]z[/katex]-axis be chosen perpendicular to the plane containing vector [katex] \overrightarrow a[/katex] and vector [katex] \overrightarrow b[/katex]. Then, we have
Important Note :
(3) In [katex] \left( \overrightarrow a\times\overrightarrow b\right)\times\overrightarrow c[/katex], suppose the vectors inside the brackets, be called vector [katex] \overrightarrow b[/katex] as the middle vector and vector [katex] \overrightarrow a[/katex] as the non-middle vector. Similarly, in [katex] \overrightarrow a\times\left(\overrightarrow b\times\overrightarrow c\right)[/katex] , vector [katex] \overrightarrow b[/katex] is the middle vector and vector [katex] \overrightarrow c[/katex] 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.