Question

Sketch the curve represented by the vectorvalued function and give the orientation of the curve. \(\mathbf{r}(\theta)=\cos \theta \mathbf{i}+3 \sin \theta \mathbf{j}\)

Short answer

The curve represented by the function \( \mathbf{r}(\theta) = \cos \theta \mathbf{i} + 3\sin \theta \mathbf{j} \) is an ellipse oriented in counter-clockwise direction.

Step-by-step solution

- Conversion to Rectangular Coordinates

To graph this function, it is helpful to convert it to rectangular coordinates. Rectangular coordinates uses the notation (x,y) where \( x = \cos \theta \) and \( y = 3\sin \theta \). Thus, we can rewrite \( \mathbf{r}(\theta) \) as \((\cos \theta, 3\sin \theta)\)

- Sketching the Curve

Next, sketch this curve on the plane. This can be done by plotting several points. For instance, for \( \theta = 0 \), \( \pi/2 \), \( \pi \), \( 3\pi/2 \) and \( 2\pi \), the points (1,0), (0,3), (-1,0), (0,-3) and (1,0) are obtained, respectively. Linking these points, it is observed that this represents an ellipse with semi-major axis of length 3 in the y-direction and semi-minor axis of length 1 in the x-direction. Set the orientation to be counter clockwise, as \( \theta \) increases from 0 to \( 2\pi \).

- Define the Orientation

The orientation is counter-clockwise. The vector function \( \mathbf{r}(\theta) \) starts from the point (1,0) at \( \theta = 0 \), moves counter-clockwise to the point (0,3) at \( \theta = \pi/2 \), continues to the point (-1,0) at \( \theta = \pi \), to the point (0,-3) at \( \theta = 3\pi/2 \), and finally returns to the starting point at \( \theta = 2\pi \). This movement is counter-clockwise around the ellipse.

Key concepts

Parametric Equations

Parametric equations represent a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. For example, the equations of a curve in the plane usually involve two variables: one for the horizontal (the x-coordinate) and one for the vertical (the y-coordinate).

In the context of the given exercise, the vector-valued function \( \mathbf{r}(\theta)=\cos \theta \mathbf{i}+3 \sin \theta \mathbf{j}\) defines a curve in two dimensions. Each value of the parameter \(\theta\) corresponds to a point on the curve, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along the x and y axes, respectively. This type of representation is very useful for describing complex motions and paths in physics and engineering.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, provide a method of rendering points in a plane using two perpendicular coordinate axes. In this system, any point in the plane can be specified by an x-coordinate and a y-coordinate, which represent the point's horizontal and vertical positions, respectively.

When we convert the parametric equations into rectangular coordinates, we replace the parameter \(\theta\) with variables x and y. In the exercise, x is defined as \(x = \cos \theta\) and y as \(y = 3\sin \theta\). This transformation makes it easier to recognize the shape of the curve as an ellipse and to plot it using familiar x and y values.

Sketching Curves

Sketching curves involves drawing the path a curve takes through the plane based on the equations that define it. When working with vector-valued functions or parametric equations, plotting several points determined by varying the parameter can help visualize the curve's overall shape.

For the exercise at hand, points are plotted using selected values of \(\theta\) to provide a frame of reference. As indicated in the step-by-step solution, plotting points like (1,0), (0,3), (-1,0), and (0,-3) for corresponding \(\theta\) values of 0, \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\), respectively, and then connecting these points with a smooth curve, reveals the elliptical shape governed by the function \( \mathbf{r}(\theta)\).

Ellipse Orientation

The orientation of an ellipse in a parametric curve refers to the direction in which it is traversed as the parameter increases. In the case of \( \mathbf{r}(\theta)\), as \(\theta\) grows from 0 to \(2\pi\), the orientation determined is counter-clockwise.

Understanding the orientation is crucial for various applications, including physics and engineering problems where the direction of motion is significant. By following the path from (1,0) at \(\theta = 0\) through the subsequent points and back to the start, we affirm the ellipse's counter-clockwise orientation, as described in the original exercise solution. This orientation is a property of the way the ellipse is being traced by the parametric equations.