If a =a₁ i +a₂ j +a₃ k , b =b₁ i +b₂ j +b₃ k , c = c ₁ i + c ₂ j + c ₃ k and [ array lll 3 a + b & 3 ~b + c & 3 c + a array ]= | array lll a i & a j & a k b i & b j & b k c i & c j & c k array | then the value of is

Step-by-step solution

| array lll a i & a j & a k b i & b j & b k c i & c j & c k array |= | array lll a ₁ & a ₂ & a ₃ ~b ₁ & ~b ₂ & ~b ₃ c ₁ & c ₂ & c ₃ array |= [ array lll a & b & c array ] [ array lll 3 a + b & 3 ~b + c & 3 c + a array ] aligned & =(3 a + b ) ((3 ~b + c ) (3 c + a )) & =(3 a + b ) (9( ~b c )+3( ~b a )+3( c c )+( c a )) aligned aligned & =(3 a + b ) (9( ~b c )+3( ~b a )+0+( c a )) & =27 a ( b c )+0+0+0+0+ b ( c a ) & =27 [ array lll a & b & c array ]+ [ array lll b & c & a array ] & =27 [ array lll a & b & c array ]+