The line x-2 3 = y-3 4 = z-4 5 is parallel to the plane

Step-by-step solution

The direction vector of the line x-2 3 = y-3 4 = z-4 5 is v = 3 i + 4 j + 5 k . A line is parallel to a plane if its direction vector v is perpendicular to the normal vector n of the plane, which implies the dot product v n = 0 . Checking the options: For option (1), the normal vector is n = 3 i + 4 j + 5 k . The dot product is v n = (3)(3) + (4)(4) + (5)(5) = 9 + 16 + 25 = 50 0 . For option (2), the normal vector is n = 2 i + 3 j + 4 k . The dot product is v n = (3)(2) + (4)(3) + (5)(4) = 6 + 12 + 20 = 38 0 . For