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# Tangent Vector

For a curve with radius vector , the unit tangent vector is defined by

 (1) (2) (3)

where is a parameterization variable, is the arc length, and an overdot denotes a derivative with respect to , . For a function given parametrically by , the tangent vector relative to the point is therefore given by

 (4) (5)

To actually place the vector tangent to the curve, it must be displaced by . It is also true that

 (6) (7) (8)

where is the normal vector, is the curvature, is the torsion, and is the scalar triple product.

Binormal Vector, Curvature, Manifold Tangent Vector, Normal Vector, Tangent, Tangent Bundle, Tangent Plane, Tangent Space, Torsion Explore this topic in the MathWorld classroom

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## References

Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 108-111, 1997.

Tangent Vector

## Cite this as:

Weisstein, Eric W. "Tangent Vector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TangentVector.html