Vector Triple Product

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.


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.

Leave a Reply

Your email address will not be published. Required fields are marked *