Let the equations of three planes be P₁: 2x + y - z = 3 P₂: x + y + z = 4 P₃: x - 2y + z = If these three planes intersect to form a triangular prism (i.e., they are mutually parallel to one line but do not share a common line), then the correct conditions on and are

Step-by-step solution

For the three planes to form a triangular prism, the corresponding system of linear equations must have no solution. This requires the determinant of the coefficient matrix to be zero: = vmatrix 2 & 1 & -1 1 & 1 & 1 1 & -2 & vmatrix = 0 Expanding along the first row: 2( + 2) - 1( - 1) - 1(-2 - 1) = 0 2 + 4 - + 1 + 3 = 0 = -8 Now, we check for consistency by finding a linear combination of the first two equations that matches the LHS of the third equation. Let a(2x + y - z) + b(x + y + z) = x - 2y - 8z Comparing coe