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COMEDK2022MathematicsThree Dimensional Geometry

The line x-3 4 = y-4 5 = z-5 6 is parallel to the plane

Options

  1. A3 x+4 y+5 z=7
  2. Bx+y+z=2
  3. Cx-2 y+z=0
  4. D2 x+3 y+4 z=0

Correct answer

C. x-2 y+z=0

Step-by-step solution

The direction vector of the given line x-3 4 = y-4 5 = z-5 6 is v = 4 i + 5 j + 6 k . A line is parallel to a plane if its direction vector v is perpendicular to the normal vector n of the plane, which means the dot product v n = 0 . Checking the options: For (7XqPP): n = 3 i + 4 j + 5 k . v n = (4)(3) + (5)(4) + (6)(5) = 12 + 20 + 30 = 62 0 . For (L48YE): n = 1 i + 1 j + 1 k . v n = (4)(1) + (5)(1) + (6)(1) = 4 + 5 + 6 = 15 0 . For (2Ytuu): n = 1 i - 2 j + 1 k . v n = (4)(1) + (5)(-2) + (6)(1) = 4 - 10 + 6 = 0 . S

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