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Determine a Cartesian equation for given planes

by RoRi

For each of the following planes, find a linear Cartesian equation of the form

    \[ ax + by + cz = d \]

that describes the plane.

  1. The plane through (2,3,1) spanned by (3,2,1), \ (-1,-2,-3).
  2. The plane through the points (2,3,1), \ (-2,-1,-3), \ (4,3,-1).
  3. The plane through the point (2,3,1) parallel to the plan through (0,0,0) spanned by (2,0,-2) and (1,1,1).
  1. The plane through (2,3,1) spanned by (3,2,1) and (-1,-2,-3) is the set of points

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

    Therefore, we have the parametric equations

        \[ x = 2 + 3s - t, \qquad y = 3+2s-2t, \qquad z = 1+s-3t. \]

    Then we want to solve for s,t in terms of x,y,z. From the first equation we have

        \[ x = 2+3s-t \quad \implies \quad t = 2+3s - x. \]

    From the second equation we then have

        \[ y = 3+2s - 4 - 6s + 2x \quad \implies \quad s = \frac{-1 + 2x - y}{4}. \]

    Which gives us

        \[ t = 2 + \frac{-3 + 6x - 3y}{4} - x. \]

    So, from the third equation we then have

        \begin{align*} z &= 1 + \frac{-1+2x-y}{4} - 6 + \frac{9-18x+9y}{4} + 3x \\ &= -5 + \frac{8-16x+8y}{4} + 3x \\ &= -3-x+2y. \end{align*}

    Thus,

        \[ z = -3-x+2y \quad \implies \quad x - 2y + z = -3 \]

    is the requested linear Cartesian equation.

  2. The plane through the three points (2,3,1), \ (-2,-1,-3), \ (4,3,-1) is the set of points

        \[ M = \{ (2,3,1) + s(-4,-4,-4) + t(2,0,-2) \}. \]

    But, (-4, -4, -4) and (2,0,-2) are in the linear span of (3,2,1), \ (-1,-2,-3) since

        \begin{align*} (-4,-4,-4) &= (-1,-2,-3) - (3,2,1) \\ (2,0,-2) &= (3,2,1) + (-1,-2,-3). \end{align*}

    Thus, this plane is equal to the plane in part (a). Hence, we have the linear Cartesian equation,

        \[ x - 2y + z = -3. \]

  3. Again, this is the same plane as in parts (a) and (b) since the span of (2,0,-2) and (1,1,1) is the same as the span of (3,2,1) and (-1,-2,-3). Hence, the requested linear Cartesian equation is

        \[ x - 2y + z = -3. \]

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