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Determine parametric equations for given planes

by RoRi

Determine scalar parametric equations for the following planes.

  1. The plane through (1,2,1) spanned by (0,1,0) and (1,1,4).
  2. The plane through (1,2,1), \ (0,1,1), \ (1,1,4).
  1. The plane through (1,2,1) spanned by (0,1,0) and (1,1,4) is the set of points

        \[ M = \{ (1,2,1) + s(0,1,0) + t(1,1,4) \} = \{ (1+t, 2+s+t, 1+4t) \}. \]

    Therefore, we have the scalar parametric equation

        \[ x = 1+t, \qquad y = 2+s+t, \qquad z = 1+4t. \]

  2. The plane through (1,2,1), \ (0,1,0), \ (1,1,4) is the set of points

        \[ M = \{ (0,1,0) + s(1,1,1) + t(1,0,4) \} = \{ (s+t, 1+s, s+4t) \}. \]

    Therefore, we have the scalar parametric equation

        \[ x = s+t, \qquad y = 1+s, \qquad z = s + 4t. \]

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