Distance Between Parallel Planes

Question

Over the past few days, I have been learning about the distance between a point and a plane and the distance between two parallel planes. I am currently having some trouble in understanding the latter.

I have been learning from this video here - https://www.youtube.com/watch?v=Tl6x2u88c1w, as well as others which I have found tougher to understand.

We are given the plane equations:

$2x - 3y + z = 4$

$4x - 6y + 2z = 3$

At around 3:49, the video presenter says that we are going to find a point on one of the planes, and that the way to do this is through setting the $y$ and $z$ coefficients to $0$. The objective is to find the shortest distance between this point and the other plane.

However, the presenter unfortunately doesn't take the time to explain the purpose of setting the $x$ and $y$ coefficients to $0$. Nonetheless, this would turn the first plane equation into:

$2x = 4$

And from here it is easy to see that $x = 2$. Which means that the coordinates of the point on the plane are:

$\{2, 0, 0\}$

At this point; I must ask:

How can we know/guarantee that this point is on the plane? It seems to me that just replacing $x$ and $y$ with $0$ provides no guarantee that this point will indeed lie on the plane. If prospective respondents are able to provide some kind of proof or explanation for how we know this then it would be greatly appreciated.

I do have further questions on this topic, but seeing as though this post is already fairly long, I will save them for a separate post.

Thanks in advance.

Answer 1

Convert your second equation of the plane into the Hessian normalform: $$\frac{4x-6y+2z-3}{\pm\sqrt{16+36+4}}=0$$ and plug in the point situated on the first plane.

Answer 2

Does point $(2,0,0)$ satisfy the plane equation? By setting $y=0, z=0$ we are essentially finding a point of intersection between the plane and axis $x$. Yes, it's possible that our plane is parallel to axis x but in that case $x$ would not be part of the plane equation. Please note that we set $y=0, z=0$ just to make is easier to find $x$, we can set any two coordinates to any values and that will give us an equation to find the third one.