finding a line passing through two points and a plane perpendicular to that point and passing through a third solution verification

Question

I have the points $A(2,0,0), B(2,4,-3), M(-4,0,0)$

I have to find a line g passing through the points A and B, and a plane which is perpendicular to g and it's passing through M.

So my approach is the following:

For the coordinates of g I just use the coordinates of AB which are $(2-2, 4-0,-3-0)=(0,4,-3).$

And for $\alpha$ I use the fact that when the line g is perpendicular to $\alpha$ their representing vectors are going to have zero scalar product: 0x + 4y-3z+D=0

Substituting M coordinates I get that D = 0, and from there $\alpha:0x+4y-3z=0$

Is this an alright solution and is my uderstanding for the process correct?