The relation (a xx b)*(c xx d)=(a*c)(b*d)−(a*d)(b*c) (5.11) is called the Identity of Langrange. From (5.11) follows (a xx b)xx c=(a*c)b−(b*c)a (5.12) Can someone give a hint?

Patience Owens

Patience Owens

Open question

2022-08-20

How can one obtain (5.12) from (5.11)?
The relation
(5.11) ( a × b ) ( c × d ) = ( a c ) ( b d ) ( a d ) ( b c )
is called the Identity of Langrange. From (5.11) follows
(5.12) ( a × b ) × c = ( a c ) b ( b c ) a
Can someone give a hint?

Answer & Explanation

contidt9869d3

contidt9869d3

Beginner2022-08-21Added 7 answers

A key observation is that if A and B are two vectors such that A C = B C for all vectors C , then A = B . This fact will be used here.
If d is an arbitrary vector, then
[ ( a × b ) × c ] d = ( a × b ) ( c × d )
Use (5.11) to obtain the equivalent expression
[ ( a c ) b ( b c ) a ] d
Since d was arbitrary, we deduce (5.12).

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