Normal Lines and Tangent Planes

NORMAL LINES AND TANGENT PLANES

Given a surface defined by F(x,y,z) = 0, would there be a vector orthogonal to every curve on the surface passing through a given point on the surface, say point A(x₁, y₁, z₁)? For the vector to be orthogonal to a curve at A it would have to be orthogonal to a vector tangent to the curve at A. Click here or on the picture to see an enlargement.

be any curve on the surface passing through A at t = t₁.

so by the chain rule

Click here to see an enlargement that includes the tangent plane. Here is a DPGraphPicture of the surface and the tangent plane. Here is an enlargement of the picture above that includes the tangent lines. And finally, here is an enlargement of the picture at the right with a normal vector added in red.

Here is a link to a video presentation with audio of me describing how to find a tangent plane and normal line to the surface shown here.

return to exam III notes and links

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu