Tangent Planes and Normal Lines

Definition (Tangent Planes and Normal Lines) Suppose the surface has a nonzero normal vector at the point Then the line through parallel to is called the normal line to at and the plane through with normal vector is the tangent plane to at

Proposition (Tangent Planes and Normal Lines)  Suppose is a surface with the equation and let be a point on where is differentiable with Then the equation of the tangent plane to at is  



and the normal line to at has parametric equations





and


If we have and so and and the equation of the tangent plane becomes  

Example (Tangent Planes and Normal Lines) Find the equation of the tangent plane.

(a) Find the equations for the tangent plane and the normal line to the cone at the point where and

    Solution. If is the point of tangency and and then If we consider then the cone can be regarded as the level surface The partial derivatives of are , , and   so at , and Thus the tangent plane has the equation



or and the normal line is given parametrically by the equations and

(b) Find the equations of the tangent plane and the normal line at the point to the ellipsoid

    Solution. The ellipsoid is a level surface of the function Therefore, we have  





Then the tangent plane at is



which simplifies to and the parametric equations for the normal line are: and    

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Tangent Planes And Normal Lines
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/tangent-planes-and-normal-lines.html