Question

Sketch the graph of the plane curve given by the vector-valued function and, at the point on the curve determined by \(\mathbf{r}\left(t_{0}\right),\) sketch the vectors \(\mathbf{T}\) and \(\mathbf{N}\). Note that \(\mathbf{N}\) points toward the concave side of the curve. $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j} \quad t_{0}=\pi $$

Short answer

The plane curve represented by vector-valued function is an ellipse with semi-axes of lengths 3 and 2. At the point \(t_{0}=\pi\), the unit tangent vector \(\mathbf{T}\) and the principal unit normal vector \(\mathbf{N}\) are computed and sketched. The tangent vector \(\mathbf{T}\) is tangent to the curve, and the normal vector \(\mathbf{N}\) is perpendicular to \(\mathbf{T}\) and points to the concave side of the curve.

Step-by-step solution

Sketch the Plane Curve

Start by identifying that \(\mathbf{r}(t)\) represents a parametric equation for the curve with \(x=3\cos(t)\) and \(y=2\sin(t)\). This is an ellipse with semi-axes of lengths 3 units (on the x-axis) and 2 units (on the y-axis). So, plot the ellipse.

Compute the Tangent Vector

Next, compute the tangent vector \(\mathbf{T}\) at the point \(t_{0}=\pi\). The unit tangent vector \(\mathbf{T}\) is the derivative of the position vector \(\mathbf{r}(t)\) divided by its magnitude. Therefore, first compute \(\mathbf{r}'(t)\) at \(t_{0}=\pi\), and then normalize it to get the unit tangent vector \(\mathbf{T}\).

Compute the Normal Vector

After that, compute the principal unit normal vector \(\mathbf{N}\). The unit normal vector \(\mathbf{N}\) is the derivative of the unit tangent vector \(\mathbf{T}\) divided by its magnitude. So, also find \(\mathbf{T}'(t)\) at \(t_{0}=\pi\), and then normalize it to get the unit normal vector \(\mathbf{N}\).

Sketch the Vectors

Finally, at the point \(\mathbf{r}(\pi)\) on the ellipse, sketch the vectors \(\mathbf{T}\) and \(\mathbf{N}\). \(\mathbf{T}\) is the unit tangent vector, so it will be tangent to the curve at that point. On the other hand, \(\mathbf{N}\), the principal unit normal vector, is perpendicular to \(\mathbf{T}\) and points to the concave side of the curve as noted in the problem statement.