Equation of a line passing trough a point and parallel to a plane

Question

I'm trying to solve this simple exercise but I don't really know how to manage it. I need to find the equation of a line passing trough $P_0=(1,2,3)$ and parallel to the plane $\pi: x-2y = 0$. I tought to start from finding a plan parallel to the $\pi$ passing trough $P_0$:

$x-2y+d=0$

$1-4+d=0, d=3$

The parallel plan passing trough $P_0$ will be $\alpha : x-2y+3=0 $

Now to find the line equation I need some way to restrict since such line will not be unique. What's the fastest method to determine the line equation?

Answer 1

Note that the plane has the normal vector $(1, -2, 0)$ and that every parallel line to the plane has a direction vector which is orthogonal to the normal vector. Take for example $(0,0,1)$, then you can choose the line with equation

$$g: x = (1, 2, 3) + \lambda\cdot (0, 0, 1),$$ whereby $\lambda \in \mathbb{R}$.