Difference between revisions of "Tangent Plane"
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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. | 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=== | + | ====Definition==== |
: Let <math> P_0 = (x_0,y_0,z_0) </math> be a point on a surface <math>S</math>, and let <math>C</math> be any curve passing through <math>P_0</math> and lying entirely in <math>S</math>. If the tangent lines to all such curves <math>C</math> at <math>P_0</math> lie in the same plane, then this plane is called the tangent plane to <math>S</math> at <math>P_0</math>. | : Let <math> P_0 = (x_0,y_0,z_0) </math> be a point on a surface <math>S</math>, and let <math>C</math> be any curve passing through <math>P_0</math> and lying entirely in <math>S</math>. If the tangent lines to all such curves <math>C</math> at <math>P_0</math> lie in the same plane, then this plane is called the tangent plane to <math>S</math> at <math>P_0</math>. | ||
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==Resources== | ==Resources== | ||
Revision as of 10:45, 6 October 2021
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 be a point on a surface , and let be any curve passing through and lying entirely in . If the tangent lines to all such curves at lie in the same plane, then this plane is called the tangent plane to at .
Resources
Videos
- 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
