The lines r = a + ( b c ) and r = c + ( a b ) will intersect if

Step-by-step solution

Given lines: L₁: r = a + ( b c ) L₂: r = c + ( a b ) For intersection, there exist , such that a + ( b c ) = c + ( a b ) . Dot both sides with b : b a + [ b ( b c )] = b c + [ b ( a b )] Since scalar triple products with repeated vectors vanish, b a = b c This is necessary for intersection. The general intersection condition is ( a ₂ - a ₁) ( d ₁ d ₂) = 0 . Here, a ₁ = a , d ₁ = b c , a ₂ = c , d ₂ = a b . Compute ( b c ) ( a b ) = -[ a b c ] b using vector triple product identity. The condition becomes: ( c - a )