Calc 3: What is the difference between a tangent set and tangent plane? (not conceptually but mathematically) The equation of a tangent SET is:
z = f(p) + (gradient vector of f(p) dot product with the displacement vector p)
The equation of a tangent PLANE is:
0 = (gradient vector of f(p) dot product with the displacement vector p)
Why are these different?
Is it because when we're finding the tangent plane, f(p) is already taken into account? For example with the following question:
F(x, y, z) = x^2 + y^2 + z
p= (1, 2, 3)
Q. find an equation for the tangent plane to the level surface at point p.
F(p) = 8
Gradient Vector:
<2x, 2y, 1>
Gradient Vector at p:
<2,4,1>
The equation is 2(x-1) + 4(y-2) + z = 0
But why is it not (using tangent set equation) 2(x-1) + 4(y-2) + z = -8 ?
I understand visually that the = 0 equation is correct.
 A: I don't think either equations you've described correctly answer the example. Furthermore "the example Q" itself is somewhat vague. The reason I say this is that there may be multiple level surfaces of F that have $\textit{different}$ tangent planes at p (not 100% sure about this, but I don't see why it wouldn't be the case).
For the function given, there is only one level surface that contains p, this is F=8. So the example makes sense to me if we restate it as:
Q. Let $F(x,y,z)=x^2+y^2+z$ and $P = (1,2,3)$. Find an equation for the plane that is tangent to the level surface of $F$ which contains the point $P$.
Also, I think you should double check the definitions at the top of your question. What do you mean by:
The tangent SET is: $z=\nabla F(p)\cdot p$
The tangent PLANE is: $\nabla F(p) \cdot p =0$
If we look at these in our example then the tangent SET is $3\times 7$ and the tangent PLANE says 13 is 0.
Can you show me mathematically, not conceptually, how you got to $2(x-1)+4(y-2)+z=0$ and $2(x-1)+4(y-2)+z=-8$?
To find the tanget plane to the level surface at p answer the example try using $\nabla F(p) \cdot (\textbf{x}-p)=0$
where $\mathbf{x}=<x,y,z>$ is some arbritrary point on the plane we're describing.
A: Tangent is a line like :
$(x-a)/l=(y-b)/m=(z-c)/n=t.$ but a plane is a linear connection of $x,y,z$ like $Ax+By+Cz=D$.
