<p>We have <span>\([\vec{a}\times\vec{b}\;\; \vec{b}\times\vec{c}\;\; \vec{c}\times\vec{a}] = \lambda[\vec{a}\vec{b}\cdot\vec{c}]^2\)</span>. Find <span>\(\lambda\)</span>.</p>
Step-by-Step Solution
Key Concept: Use the cyclic property of scalar triple product and the identity that [a×b, b×c, c×a] relates to (a·b·c)² through the determinant expansion and BAC-CAB rule applied systematically.
Step 1: Express the left side using scalar triple product notation. We need [ a × b , b × c , c × a ] = ( a × b )·[( b × c )×( c × a )] Step 2: Apply BAC-CAB rule: ( b × c )×( c × a ) = c [( b × c )· a ] - a [( b × c )· c ] = c [ a ·( b × c )] - a ·0 = [ a b c ] c Step 3: Thus ( a × b )·[( b × c )×( c × a )] = [ a b c ]( a × b )· c = [ a b c ]·[ a b c ] = [ a b c ]^2 Step 4: Since [ a b c ] = a ·( b × c ), we have [ a × b b × c c × a ] = [ a b c ]^2 Step 5: Comparing with λ[ a b · c ]^2, we get λ = 1 ∴ Answer: λ = 1
Correct Answer: B