Find parametric equation for a plane

Question

How do I write a parametric equation for a plane $m$, if $m$ is a plane through $2$ points

$$a = (1, 0, 0)$$ $$b = (2, 2, 3)$$ and parallel to the line $$C : (x, y, z) = (5 - t, 1, t)$$

I would like some help and hints about how to think and go forward. Thank you.

Answer 1

Hint:

If the plane passes through the points $a$ and $b$ it is parallel to the vector $ \overrightarrow{a b}=(-1,-2,-3)$.

If it is parallel to the given line it is parallel to the orienting vector of the line $\vec{c}=(-1,0-1)$.

So you can find a vector orthogonal to the plane $\vec{k}=\overrightarrow{a b} \times \vec{c}$ and you have a point of the plane ($a$ or $b$), so you can find the equation of the plane: $(\vec x-\vec a) \cdot \vec k=0$.