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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 :

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

Result:

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

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.

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