Tangent Plane

Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A tangent plane at a regular point contains all of the lines tangent to that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly.

Definition

Let $P_{0}=(x_{0},y_{0},z_{0})$ be a point on a surface $S$, and let $C$ be any curve passing through $P_{0}$ and lying entirely in $S$. If the tangent lines to all such curves $C$ at $P_{0}$ lie in the same plane, then this plane is called the tangent plane to $S$ at $P_{0}$.

Resources

Videos

• Tangent planes | MIT 18.02SC Multivariable Calculus, Fall 2010

• Determining the Equation of a Tangent Plane Video by Mathispower4u 2011

• Ex 1: Find the Equation of a Tangent Plane to a Surface video by Mathispower4u 2014

• How To Find The Equation of the Normal Line-The Organic Chemistry Tutor 2014 Video by The Organic Chemistry Tutor 2014

• quation of the normal line at a point (KristaKingMath) Video by KristaKingMath 2012

• Tangent Plane Approximations Video by -patrickJMT