If a = x i + y j +4 z k , ~b = y i +x j +3 y k , c =- z i -2 z j -( +1) k x are the sides of the triangle ABC , where x, y , z are not all zero, such that a + b - c = 0 , then value of is

Step-by-step solution

Given vectors a = x i + y j + 4z k , b = y i + x j + 3y k , and c = -z i - 2z j - ( +1)x k with a + b - c = 0 , we substitute and group components. The i component yields x + y + z = 0 The j component gives x + y + 2z = 0 The k component results in ( +1)x + 3y + 4z = 0 This homogeneous system has nontrivial solutions only if the coefficient matrix determinant is zero: pmatrix & 1 & 1 1 & 1 & 2 +1 & 3 & 4 pmatrix = (4-6) - 1(4-2 -2) + 1(3- -1) = -2 - 2 + 2 + 2 - = - Setting - = 0 gives the solution = 0 . Final answe