Compute Unit Normal Vector: Why Derivative is Orthogonal

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SUMMARY

The unit normal vector N of a curve is defined as the first derivative of the unit tangent vector T with respect to the parameter t, normalized by the norm of T'(t). The discussion clarifies that while T(t) is orthogonal to T'(t), the orthogonality of T'(t) to T(t) holds true only when the tangent vectors maintain a constant length. This implies that the derivative of the tangent vector is not universally orthogonal to the tangent vector itself unless specific conditions regarding the length are met.

PREREQUISITES
  • Understanding of parametric vector equations
  • Familiarity with vector calculus concepts
  • Knowledge of unit tangent vectors and their properties
  • Basic principles of derivatives in multivariable calculus
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  • Study the properties of unit tangent vectors in vector calculus
  • Learn about the implications of constant length in vector functions
  • Explore the concept of curvature and its relationship with normal vectors
  • Investigate the derivation of the Frenet-Serret formulas
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Students and professionals in mathematics, physics, and engineering who are studying vector calculus, particularly those interested in the geometric properties of curves and their derivatives.

lordkelvin
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The unit normal vector N of a given curve is equal to the first derivative with respect to t of the unit tangent vector T'(t)divided by the norm of T'(t) (For a parametric vector equation of parameter t.)

I realize this works because T(t) is orthogonal to T'(t), but I don't understand why the derivative of the vector T is orthogonal to T itself.

Can anyone explain to me why the derivative of a tangent vector is orthogonal to the tangent vector? Thanks.
 
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lordkelvin said:
Can anyone explain to me why the derivative of a tangent vector is orthogonal to the tangent vector? Thanks.

In general it is not true but if the tangent vectors have constant length then the derivative of the length is zero.
 

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