When I have a plane, I can calculate two equations:

**Parametric**, of the form
$$P + s\cdot \vec{q} + t \cdot \vec{r}$$
For some point $P$ in the plane, two direction arrows $\vec{q},\vec{r}$ that are not parallel, and some $s,r\in \mathbb{R}$.

**Normal**, of the form

$$ax+by+cz = d$$
With only one point and one direction arrow perpendicular to the plane.

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I am wondering, is it possible, given one form, to transform it into the other?

Here I have two planes:

$$A: x - y + 2z = 10\\
B: (1,1,1)+t(1,-1,1)+s(0,1,0)$$

$A$ is normal and $B$ is parametric. How can I transform $A$ to parametric and $B$ to normal?

I happen to know that a normal form of $B$ is

$$-x+z=0$$

From a previous exercise, but I'm not sure what was done to achieve this form.