Let σ 1 ,   σ 2 ,   σ 3 be planes passing through the origin. Assume that σ 1 is perpendicular to the vector 1 , 1 , 1 , σ 2 is perpendicular to a vector a , b , c and σ 3 is perpendicular to the vector a 2 , b 2 , c 2 . What are all the positive values of a ,   b and c so that σ 1 ∩ σ 2 ∩ σ 3 is a single point?

Step-by-step solution

According to given information equation of planes σ 1 ; x + y + z = d 1 σ 2 : a x + b y + c z = d 2 and σ 3 : a 2 x + b 2 y + c 2 z = d 3 Now, for unique solution Δ ≠ 0 . and Δ = 1 1 1 a b c a 2 b 2 c 2 On applying c 2 → c 2 - c 1 and c 3 → c 3 - c 1 Δ = 1 0 0 a b - a c - a a 2 b 2 - a 2 c 2 - a 2 = b - a c - a 1 0 0 a 1 1 a 2 b + a c + a On applying c 3 → c 3 - c 2 b - a c - a 1 0 0 a 1 0 a 2 b + a c - b = a - b b - c c - a So, if a , b and c has any three dist