6.6 Jacobi’s Identity and Lagrange’s Identity

Theorem . (Jacobi’s identity) For any three vectors , , , a b c  we have + × Proof Using vector triple product expansion, we have  = ( a c b a b c  = ( b a c b c a ... ( ) ... ( ) a i a i b c i b c j b c b c k a b c k a b c b c a c b a b c a c b i b j a b c i c j c k a b c b c a b c k ´ ´ ´ ´ =- ´ Now, = ( c b a c a b Adding the above equations and using the scalar product of two vectors is commutative, we get + × Theorem .

(Lagrange’s identity) For any four vectors , , , , a b c d we have ( ) ( a c a d b c b d Proof Since dot and cross can be interchanged in a scalar product, we get ) ( )) (( ) ) b d c b c d (by vector triple product expansion) = ( )( )( a c b d a d b c = a c a d b c b d Example . Prove that ] [ , , ] b b c c a b c Using the definition of the scalar triple product, we get ]   b b c c = ( ) {( ) ( )} ... ( ) By treating (  as the first vector in the vector triple product, we find ) (  = (( (( [ , , ] a c c a a b c c  . Using this value in ( ), we get ] b b c c  = ) ([ , , ] ) [ , , ]( [ , , ] a b c c a b c a a b c Example .

Prove that ( )) ) ( c a  . Treating ( as the first vector on the right hand side of the given equation and using the vector triple product expansion, we get ) (  = (( (( )) c a a c c a  . Example . For any four vectors , a b c d we have ) ( = [ , , ] [ , , ] [ , , ] [ , , ] a b d c a b c d a c d b b c d a  .

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