Let [a xx b \ \ b xx c \ \ c xx a]=k[a b c]^2. Find k. Here, [u v w]=u * (v xx w).

Sydney Stein

Sydney Stein

Answered question

2022-08-09

Let [ a × b   b × c   c × a ] = k [ a   b   c ] 2 . Find k. Here, [ u   v   w ] = u ( v × w )
My attempt:
Writing scalar triple product as : ( a × b ) ( ( b × c ) × ( c × a ) ) . Not able to proceed next.

Answer & Explanation

Izabella Fisher

Izabella Fisher

Beginner2022-08-10Added 14 answers

As hinted in a comment, we use the identity u × ( v × w ) = ( u w ) v ( u v ) w . Let
Δ = a ( b × c ) = b ( c × a ) = c ( a × b ) .
First we compute
( b × c ) × ( c × a ) = ( ( b × c ) a ) c ( ( b × c ) c ) a = Δ c 0 a = Δ c .
Therefore
( a × b ) ( ( b × c ) × ( c × a ) ) = ( a × b ) Δ c = Δ ( ( a × b ) c ) = Δ 2 .
Hence k=1.

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