If the vectors (a i + j + k , i +b j + k ) and ( i + j +c k ) are coplanar, where ((a, b, c 1 ) ), then the value of ( 1 1-a + 1 1-b + 1 1-c = )

Step-by-step solution

For coplanar vectors, ( | array ccc a_x & a_y & a_z b_x & b_y & b_z c_x & c_y & c_z array |=0 ) Substituting values we have, ( | array lll a & 1 & 1 1 & b & 1 1 & 1 & c array |=0 ) (R₂ R₂-R₁ and R₃ R₃ R₁ ) ( | 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 ) ( ) Dividing by ((1-a)(1-b)(1-c) ), we get ( aligned a 1-a + 1 1-b + 1 1-c & =0 1 1-a + 1 1-b + 1 1-c & = 1 1-a - a 1-a =1 aligned )