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# Jacobi’s Identity and Lagrange’s Identity

Jacobi’s Identity Theorem and Lagrange’s Identity Theorem : Definition, Theorem, Proof, Solved Example Problems, Solution

Jacobi’s Identity and Lagrange’s Identity

Theorem 6.9 (Jacobi’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 (Lagrange’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 ( × ) 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 ( × ) 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 = ( × )  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 : Jacobi’s Identity and Lagrange’s Identity | Definition, Theorem, Proof, Solved Example Problems, Solution