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 :

Vector Triple Product

Employing the well-known properties of the vector product, we get the following theorems.

Theorem 1 :

The vector triple product satisfies the properties which are given below.

vector triple product properties


The vector triple product 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


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.

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