# Equation of plane passing through intersection of line and plane

Find the equation of the plane passing through the intersection of line $$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{2}$$ and the plane $$x-y+z=5$$

and parallel to a vector with direction ratios $<2,3,-2>$

Now point of intersection of given plane and given line is $(2,-1,2)$ and direction normal of required plane will be perpendicular to $<2,3,-2>$ but how would I get a unique equation of required plane?

• Why would you you expect a "unique" such equation? THere are infinite planes through that intersection and parallel to the given vector. – DonAntonio Mar 7 '16 at 16:14
• @Joanpemo So we will get infinite planes under given information? – Mathematics Mar 7 '16 at 16:17
• Yes. After all, if you find such plane then you can spin it around that point in the desired direction and that way you get infinite planes. – DonAntonio Mar 7 '16 at 16:30

So you want a plane through $\;(2,-1,2)\;$ and parallel to $\;(2,3,-2)\;$ , so for any $\;(a,b,c)\in\Bbb R^3\;$ the following fulfills the conditions:
$$(2,-1,2)+t(2,3,-2)+s(a,b,c)\;,\;\;t,s\in\Bbb R\;$$