- #1
autodidude
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In Stewart's calculus text, the way he derives the tangent plane equation at some point is to divide the general plane equation
[tex]a(x-x_0)+b(y-y_0)+c(z-z_0)=0[/tex] by c
This must mean c is always non-zero right? But isn't c is the 'z'-component of the normal vector to the surface at some point? If it's non-zero, does that mean that for some surface in 3D, the normal vector always has a component in the z-direction?
The one counter-example I can think of is the case of a sphere of radius 1. At (1,0,0), wouldn't the normal vector be pointing in just the x-direction? But then I'm also not sure that the derivative of z with respect to x is defined at this point...
[tex]a(x-x_0)+b(y-y_0)+c(z-z_0)=0[/tex] by c
This must mean c is always non-zero right? But isn't c is the 'z'-component of the normal vector to the surface at some point? If it's non-zero, does that mean that for some surface in 3D, the normal vector always has a component in the z-direction?
The one counter-example I can think of is the case of a sphere of radius 1. At (1,0,0), wouldn't the normal vector be pointing in just the x-direction? But then I'm also not sure that the derivative of z with respect to x is defined at this point...