How Do General Plane Equations Differ from Tangent Surface Equations?

[Calpalned]

Homework Statement

What's the difference between the two equations for a plane?

This question is somewhat related to my other, overarching question here: https://www.physicsforums.com/threads/i-am-confused-about-how-multivariable-calc-works.798798/

Homework Equations

$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
and
$z - z_0 = f_x (x_0, y_0)(x-x_0) + f_y (x_0, y_0)(y-y_0)$

The Attempt at a Solution

I'm not sure what the relationship between these two equations are. Thanks everyone.

 

[Dick]

Homework Statement

What's the difference between the two equations for a plane?

Homework Equations

$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
and
$z - z_0 = f_x (x_0, y_0)(x-x_0) + f_y (x_0, y_0)(y-y_0)$

The Attempt at a Solution

I'm not sure what the relationship between these two equations are. Thanks everyone.

There's no deep difference. The first is the general form of a plane with $(a,b,c)$ as a normal vector. The second is a specific example of a plane corresponding to a tangent surface with normal vector $(f_x,f_y,-1)$.