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Find a Cartesian equation for a plane through a point containing a given line

Let L be the line through the point (1,2,3) and parallel to the vector (1,1,1) and let (2,3,5) be a point not on L. Find a Cartesian equation for the plane M which passes through (2,3,5) and entirely contains L.


The line L is the set of points

    \[ L = \{ (1,2,3) + t(1,1,1) \} \quad \implies \quad P = (1,2,3), \ Q = (2,3,4) \in L. \]

Then, the plane M is the set of points

    \begin{align*}  M &= \{ (2,3,5) + s((1,2,3) - (2,3,5)) + t((2,3,4) - (2,3,5)) \} \\  &= \{ (2,3,5) + s(-1,-1,-2) + t(0,0,-1) \}. \end{align*}

Then, to get the Cartesian equation, we have

    \begin{align*}  x &= 2-s \\  y &= 3-s \\  z &= 5 - 2s - t. \end{align*}

The first two equations give s = 2-x and so y = 3-2 + x. This implies x - y = -1, and z is arbitrary. So, the plane is described by the equation x-y = -1.

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