Chapter 2: Space Curves
Section 2.3: Tangent Vectors
If Rs is the position vector for C, a curve parametrized by s, the arc length, and Ts=R′s is the unit tangent vector along C, show that T·T′=0, thereby proving that the unit tangent vector is necessarily orthogonal to its derivative.
Since T is a unit vector, T·T=1. Differentiate both sides to obtain T·T′=0, or
from which it follows that T·T′=0. (Note the use of the product rule for the differentiation of a dot product, as stated in Table 1.3.1.)
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