What is the normal form for this line?

Question

I have calculated the parametric form of a line as: $L = P_1 + tP_1P_3 = <2,2,0> + t<1,2,2>$.

If I am given a point $ K = <1,-1,-1>$, how would I show the normal form of plane $E$ that has the line $L$ as its normal and contains the point $K$?

Answer 1

Hint: Find a vector that is perpendicular to the direction vector of the line $<1,2,2>$. See here for the technique to find such a vector. Since you have a point $<1,-1,-1>$ and the normal to the plane $n$, then the equation of the plane is given by

$$ n.(<x,y,z>-<1,-1,-1>)=0 .$$