If the vectors a i + j + k , i +b j + k , i + j +c k (a b, c 1) are coplanar, then 1 1-a + 1 1-b + 1 1-c has the value

Step-by-step solution

Since | array lll a & 1 & 1 1 & b & 1 1 & 1 & c array |=0 Applying R ₂ R ₂- R ₁ and R ₃ R ₃- R ₁ , we get aligned & | array ccc a & 1 & 1 1-a & b-1 & 0 1-a & 0 & c-1 array |=0 & a(b-1)(c-1)-(1-a)(c-1)-(1-a)(b-1)=0 aligned Dividing by (1-a)(1-b)(1-c) , we get a 1-a + 1 1-b + 1 1-c =0 Consider, 1 1- a + 1 1- b + 1 1- c = 1 1-a - a 1-a ....[From (i)] =1