Finding the Principal Unit Normal Vector

Find the principal unit normal vector \(\mathbf{N}(t)\) to the circle \(\mathbf{r}( t) =R\;\cos t\mathbf{i}+R\;\sin t\mathbf{j}\), \(0\leq t\leq 2\pi .\) (Refer to Example 2.) Graph \(\mathbf{r}=\mathbf{r}( t), \) and show a unit tangent vector and a unit normal vector.

Solution From Example 2, the unit tangent vector \(\mathbf{T}(t)\) is \begin{equation*} \mathbf{T}(t)=-\!\sin t\mathbf{i}+\cos t\mathbf{j} \end{equation*}

Since \(\mathbf{T}^{\prime} ( t) =-\cos t\mathbf{i}-\sin t\mathbf{j},\) the principal unit normal vector \(\mathbf{N}(t)\) is \begin{equation*} \mathbf{N}(t)=\frac{\mathbf{T}^{\prime} (t)}{\left\Vert \mathbf{T}^{\prime} (t)\right\Vert }=\frac{-\!\cos t\mathbf{i}-\sin t\mathbf{j}}{\sqrt{(-\!\cos t)^{2}+(-\!\sin t)^{2}}}=-\cos t\mathbf{i}-\sin t\mathbf{j} \end{equation*}

Since \(\mathbf{r}( t) = R ( \cos t\mathbf{i}+\sin t\mathbf{j}), \) the vector \(\mathbf{N}( t)\) is parallel to \(-\mathbf{r}(t)\). That is, \(\mathbf{N}(t)\) is a unit vector opposite in direction to the vector \(\mathbf{r}(t)\), so \(\mathbf{N}\) is directed toward the center of the circle, as shown in Figure 13.