Let v ₁ = i + j + k , v ₂ = i + 2 j + 5 k , v ₃ = i + 3 j + 9 k , and u = i + a j + a^2 k . If the vector equation x v ₁ + y v ₂ + z v ₃ = u has infinitely many solutions for the scalars x, y, z , and , are the possible values of a , then the value of ^4 + ^4 is equal to :

Step-by-step solution

The given vector equation x v ₁ + y v ₂ + z v ₃ = u can be written as a system of linear equations by equating the i , j , k components: x + y + z = 1 x + 2y + 3z = a x + 5y + 9z = a^2 For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and _x = _y = _z = 0 . Let us verify the determinant of the coefficient matrix: = vmatrix 1 & 1 & 1 1 & 2 & 3 1 & 5 & 9 vmatrix = 1(18 - 15) - 1(9 - 3) + 1(5 - 2) = 3 - 6 + 3 = 0 . Now, we set _z = 0 : _z = vmatrix 1 & 1 & 1 1 &