Compute Unit Normal Vector: Why Derivative is Orthogonal

[lordkelvin]

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.

 

[lavinia]

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.