The vectors p = i +a j +a^2 k , q = i +b j +b^2 k and r = i +c j +c^2 k are non-coplanar and | array lll a & a^2 & 1+a^3 b & b^2 & 1+b^3 c & c^2 & 1+c^3 array |=0 then the value of (a b c) is

Step-by-step solution

The determinant D is given by: D = | array lll a & a^2 & 1+a^3 b & b^2 & 1+b^3 c & c^2 & 1+c^3 array | = 0 Expressing the third column as a sum of two columns yields: D = | array lll a & a^2 & 1 b & b^2 & 1 c & c^2 & 1 array | + | array lll a & a^2 & a^3 b & b^2 & b^3 c & c^2 & c^3 array | = 0 The first determinant is a Vandermonde determinant: | array lll 1 & a & a^2 1 & b & b^2 1 & c & c^2 array | = (a-b)(b-c)(c-a) Factoring a , b , and c from the rows of the second determinant gives: abc | array lll 1 & a & a^2