[Math] Find parametric equations of the plane that is parallel to the plane 3x + 2y – z = 1 and passes through the point P(l, 1, 1).

Question

Find parametric equations of the plane that is parallel to
the plane $$3x + 2y – z = 1$$ and passes through the point
$$P(1, 1, 1)$$.

How would one approach this problem?

EDIT:

The normal vector is $$(3,2,-1)$$ So I was able to get two vectors
$$(0,1,-1) $$
And
$$(2,0,-2)$$

But what would be the translation of the two vectors?

Answer 1

For two parallel planes, the normal vector is th same, so the new plane has normal vector $\vec{n}=(3,2,-1)$, and plane have to passes through the point $$P(1, 1, 1)$$ then the equation of the plane is $$3(x-1)+2(y-1)-(z-1)=0$$ and