Chapter 2: Space Curves

Section 2.3: Tangent Vectors

Example 2.3.5

$$ If $\mathbf{R}\left(s\right)$ is the position vector for $C$, a curve parametrized by $s$, the arc length, and $\mathbf{T}\left(s\right)\=\mathbf{R}\prime \left(s\right)$ is the unit tangent vector along $C$, show that $\mathbf{T}\mathbf{·}\mathbf{T}\prime \=0$, thereby proving that the unit tangent vector is necessarily orthogonal to its derivative. $$ Solution $$ Since T is a unit vector, $\mathbf{T}·\mathbf{T}\=1$. Differentiate both sides to obtain $\left(\mathbf{T}·\mathbf{T}\right)\prime \=0$, or   $\mathbf{T}·\mathbf{T}\prime \+\mathbf{T}\prime ·\mathbf{T}\=2 \mathbf{T}·\mathbf{T}\prime \=0$   from which it follows that $\mathbf{T}·\mathbf{T}\prime \=0$. (Note the use of the product rule for the differentiation of a dot product, as stated in Table 1.3.1.) $$

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