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Find the Cartesian equation of a plane given three points on the plane

by RoRi

Let (1,2,3), (2,3,4), and (-1,7,-2) be three points on a plane. Find the Cartesian equation for the plane.

First, letting P = (-1,7,-2), Q = (1,2,3), and R = (2,3,4) we compute a normal to the plane

    \begin{align*} N &= (P - Q) \times (P - R) \\ &= ((-1,7,-2) - (1,2,3)) \times ((-1,7,-2) - (2,3,4)) \\ &= (-2,5,-5) \times (-3,4,-6) \\ &= (-10,3,7). \end{align*}

So the Cartesian equation of the plane is of the form

    \[ -10x + 3y + 7z = d. \]

Since (1,2,3) is on the plane we have d = -10 + 6 + 21 = 17. Hence, the Cartesian equation for the plane is

    \[ -10x + 3y + 7z = 17 \quad \implies \quad 10x - 3y - 7z + 17 = 0. \]

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