Jacobiâ€™s Identity Theorem and Lagrangeâ€™s Identity Theorem : Definition, Theorem, Proof, Solved Example Problems, Solution

**Jacobiâ€™s
Identity and Lagrangeâ€™s Identity**

**Theorem 6.9 (Jacobiâ€™s identity)**

For any three vectors , , , we have = .

**Proof**

Using vector triple product expansion, we have

Adding the above equations and using the scalar product of two
vectors is commutative, we get

.

**Theorem 6.10 (Lagrangeâ€™s identity)**

**Proof**

Since dot and cross can be interchanged in a scalar product, we
get

**Example 6.19**

Prove that

**Solution**

Using the definition of the scalar triple product, we get

..............(1)

By treating (Ã— ) as the first vector in the
vector triple product, we find

Using this value in (1), we get

**Example 6.20**

Prove that .

**Solution**

Treating (* *Ã— * *) as the first vector on
the right hand side of the given equation and using the vector triple product
expansion, we get

**Example 6.21**

For any four vectors , , , , we have

**Solution**

Taking = ( Ã— * *) as
a single vector and using the vector triple product expansion, we get

**Example 6.22**

State whether they are equal.

**Solution**

**Example 6.23**

**Solution (i)**

By definition,

On the other hand, we have

Therefore, from equations (1) and (2), identity (i) is verified.

The verification of identity (ii) is left as an exercise to the reader.

Tags : Definition, Theorem, Proof, Solved Example Problems, Solution , 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Jacobiâ€™s Identity and Lagrangeâ€™s Identity | Definition, Theorem, Proof, Solved Example Problems, Solution

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