vectors in 3d a level maths

I get that I have to do smth with AM which is the perpendicular of the ground

A mobile phone mast, BM, is on the side of a hill. The mast is supported by a guy rope AM which is perpendicular to the hillside.

The hillside is modelled as a plane and the mobile phone mast, BM, and
guy rope, AM, are modelled as straight lines. Relative to a fixed origin, the top of the mast has coordinates M(180, -75, 20) and the guy rope is fixed to ground at A (176, -80, 2). All units are in metres.

An engineer walks in a straight line from the point C (s, 55, -5) to the
base of mast, B(180, -75, t).
Calculate the distance that the engineer walks.

(Idk how to attach an image but BM is vertical and AM is perpendicular to the slanted ground. A, B and C are on the ground, with A at a higher altitude than B and B having a higher altitude than C)

I just don't know what to even do. I get that I have to do smth with AM which is the perpendicular of the ground

Good. So you have the perpendicular, and a point on the plane, and hence an equation of the plane is?

So using the vector AM, thecartesian equation of the ground plane should be 4x+5y+18z. Am I supposed to know what it is equal to? How do I work it out? And what does it mean?

That's part of an equation. You know a point on the plane, so if you sub it in, you'll find what it's equal to, to complete the equation.

Ah ok, I should've known. And what does it represent, if anything?
I got it equals to 340

Nothing special in this circumstance.

In general, in the forumula ax+by+cz=d, if you divide through, scale, by $\sqrt{a^2+b^2+c^2}$, then the RHS is the signed distance of the plane from the origin of your coordinate system.

Ohh, I see.

So now that I have the equation of the ground plane, do I sub in the coordinates of B and C? And work out what the unknowns s and t are? And then do I dot matrix the vectors of B and C? What does dot matrix even show? Sorry I am very confused.

Ok, thank you!

Just wondering, is there another way of finding the distance between them without using 3D Pythagoras? Because since I've started the topic, we haven't had the need to use 3D pythag or 3D trig, and this question is in a homework that sums everything up.

Btw, I got 192 to 3sf, I hope that is correct
Thanks so much!!! Enjoy your evening^^