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# Vector triple product

Vector triple product is not associative.

Vector triple product

## Definition 6.5

For a given set of three vectors   , the vector ×( × ) is called a  vector triple product.

Note

Given any three vectors  , the following are vector triple products : Using the well known properties of the vector product, we get the following theorem.

## Theorem 6.7

The vector triple product satisfies the following properties. ### Remark

Vector triple product is not associative. This means that for some vectors   .

Justification The following theorem gives a simple formula to evaluate the vector triple product.

## Theorem 6.8 (Vector Triple product expansion)

For any three vectors   we have Proof

Let us choose the coordinate axes as follows :

Let x -axis be chosen along the line of action of , y -axis be chosen in the plane passing through and parallel to , and z -axis be chosen perpendicular to the plane containing and . Then, we have Note (3) In ( × ) ×c , consider the vectors inside the brackets, call as the middle vector and as the non-middle vector. Similarly, in , is the middle vector and is the non-middle vector. Then we observe 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.

Tags : Definition, Theorem, Proof , 12th Mathematics : UNIT 6 : Applications of Vector Algebra
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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Vector triple product | Definition, Theorem, Proof