# The equation of a plane that passes through a point and contains a line.

Find an equation of the plane that passes through the point P and contains the line l.

P(1,-2,3); l:r=(t,-t,2t), -∞ < t < ∞

This problem was on a test I took, which I got wrong and our professor is letting us take home the test and make corrections. But I just cant seem to figure it out, there are no examples in the book like it or online.

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$(1,-2,3)\times(1,-1,2)$ is the normalvector of the plane (using that $0$ is there). – Berci Apr 5 '13 at 15:57

It is obvious that $P$ is not on the line, now choose two points on the line (since plane is determined with three points that are not all on the same line), let us take $t=1$ and $t=2$ which correspond to the points $Q(1,-1,2)$ and $R(2,-2,4)$. The equation of the plane can be written in the form $Ax+By+Cz+D=0$. Now solve these three equations to find $A,B,C$ (first find $D$ to obtain three equations in three unknowns, $D=0$, do you see why?):
$Ax_1+By_1+Cz_1+D=0$
$Ax_2+By_2+Cz_2+D=0$
$Ax_3+By_3+Cz_3+D=0$
Easier with $t=0$ as one point in $l$. – Berci Apr 5 '13 at 15:55
@Berci I used that fact to obtain that $D=0$ but if you choose that point in these equations then one equation breaks down into $0=0$ so it is of no use to use it when solving this system. – user67878 Apr 5 '13 at 15:57