General Equation of a line

I am trying to solve my homework problem and I have a question:

Problem: Write a general equation of a line $L_1$ and consider a point $A$ that is not on the line $L_1$. Write the equation of the plane $\alpha$ passing through the line $L_1$ and the point $A$.

So I know that the general equation of a line have the form $Ax + By + C = 0$ and I took some random numbers for $A,B$ and $C$ $$ L_1 = x + 2y -4 $$ And a point $A(1;3)$. Here comes my question How should I write an equation of the plane if my line equation is in $2D$. I also tried to write it as a Vector form or in parametric form but there is written that it must be in General Form. Can you give me some hints?

Solution 1:

HINT

Once you have a line $L_{1}(t) = P + tv$, you have a direction $v$.

In order to obtain another direction corresponding to the plane we are interested in, you can consider a point $Q\in L_{1}(t)$ and the corresponding direction $w = Q - A$.

Given that you already have two directions $v$ and $w$ (which are LI) and the point $A$, you can describe the plane passing through $L_{1}(t)$ and $A$.

Hopefully this helps !