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Determine which points are on the plane with Cartesian equation 3x – 5y + z = 9

Let M be the plane with Cartesian equation 3x - 5y + z = 9.

  1. Determine which of the points (0,-2,-1), \ (-1,-2,2), \ (3,1,-5) are on the plane.
  2. Find vectors P,A,B such that M = \{ P + sA + tB \}.

  1. The point (0,-2,-1) \in M since

        \[ 3(0) - 5(-2) + (-1) = 10 - 1 = 9. \]

    The point (-1,2,2) \in M since

        \[ 3(-1) + (-5)(-2) + 2 = -3+10+2 = 9. \]

    The point (3,1,-5) \notin M since

        \[ 3(3) + (-5)(1) + (-5) = 9 - 5 - 5 = -1 \neq 9. \]

  2. We know (0,-2,-1), \ (-1,-2,2) \in M. So, let

        \[ A = Q-P = (-1,-2,2) - (0,-2,-1) = (-1,0,3). \]

    Since R = (3,1,5) \in M let

        \[ B = R-P = (3,1,5) - (0,-2,-1) = (3,3,6). \]

    Then we have

        \[ M = \{ (0,-2,-1) + s(-1,0,3) + t(3,3,6) \}. \]

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