Tangent Vectors

     We expect the tangent vector at to be the limit vector which is the vector derivative. To find parametric equations of the tangent line at we use the point corresponding to and the tangent vector



to obtain parametric equations

Proposition (Tangent Vector) Suppose is differentiable at and that Then is a tangent vector to the graph of at the point where and points in the direction of increasing .

    Proof. Let be a number in the domain of the vector function , and let be the point on the graph of that corresponds to . Then for any positive number , the difference quotient is a vector that points in the same direction as the secant vector where is the point on the graph of that corresponds to . Suppose the difference quotient has a limit as and that Then, as , the direction of the secant vector, , and hence that of the difference quotient , will approach the direction of the tangent vector of . Thus we expect the tangent vector at to be the limit vector   which is the vector derivative

Example (Tangent Vector) Find a tangent vector at the point where for



    Solution. We have,



and the tangent line to the graph of for is the line that passes through the point and is determined by the parametric equations    and because this line passes through and is parallel to the tangent vector at namely,

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Cite this as:
Tangent Vectors
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/tangent-vectors.html