Question

Sketch the vector-valued function on the given interval. $$ \vec{r}(t)=\langle 3 \sin (\pi t), 2 \cos (\pi t)\rangle, \text { on }[0,2] $$

Short answer

The function describes an ellipse traced by \( \vec{r}(t) \) on \([0,2]\).

Step-by-step solution

Understand the Function

The vector-valued function \( \vec{r}(t) = \langle 3 \sin(\pi t), 2 \cos(\pi t) \rangle \) describes a parametric curve in the plane. The parameter \( t \) varies from 0 to 2.

Determine Key Points

Evaluate the vector function at key values of \( t \) to understand the curve's behavior. Find \( \vec{r}(0), \vec{r}(1), \vec{r}(2) \). Each of these key values represents specific points on the curve.

Calculate \(\vec{r}(0)\)

Substitute \( t = 0 \) into the function: \( \vec{r}(0) = \langle 3 \sin(0), 2 \cos(0) \rangle = \langle 0, 2 \rangle \). This is the starting point of the curve.

Calculate \(\vec{r}(1)\)

Substitute \( t = 1 \) into the function: \( \vec{r}(1) = \langle 3 \sin(\pi), 2 \cos(\pi) \rangle = \langle 0, -2 \rangle \). This point is vertically below the starting point.

Calculate \(\vec{r}(2)\)

Substitute \( t = 2 \) into the function: \( \vec{r}(2) = \langle 3 \sin(2\pi), 2 \cos(2\pi) \rangle = \langle 0, 2 \rangle \). This shows that the curve returns to the starting point, completing a cycle.

Sketch the Curve

Plot the calculated points \( \langle 0, 2 \rangle \), \( \langle 0, -2 \rangle \) on a graph, and visualize the transition between these points using the periodic nature of sine and cosine, which completes an ellipse. The range of \( \vec{r}(t) \) will form an ellipse centered at the origin with semi-major axis 3 on the X-axis and semi-minor axis 2 on the Y-axis.

Key concepts

Parametric Curves

Parametric curves are a way to represent curves using parameters rather than traditional Cartesian coordinates. Here, each coordinate of the curve is expressed as a function of one variable, usually denoted as \( t \). This approach is flexible because it allows us to describe more complex shapes that might not be easily represented by simple algebraic equations.

In the case of the vector-valued function \( \vec{r}(t)=\langle 3 \sin (\pi t), 2 \cos (\pi t)\rangle \), both the \( x \)-coordinate and \( y \)-coordinate are controlled by the same parameter \( t \) as it varies from 0 to 2.

• The sine function affects the \( x \)-position while the cosine function affects the \( y \)-position.
• This description lays out a path or trajectory traced out as \( t \) changes.

Understanding how parametric curves work is vital to visualizing and sketching the path outlined by vector-valued functions.

Ellipses

An ellipse is a type of conic section that forms when a plane intersects a cone at an angle. It's a symmetrical shape resembling an elongated circle.

With the vector-valued function given, \( \vec{r}(t)=\langle 3 \sin (\pi t), 2 \cos (\pi t)\rangle \), we can identify the figure described in the plane as an ellipse. Why is this so?

• The coefficients in front of the sine and cosine functions determine the axes lengths of the ellipse.
• The coefficient 3 in \( 3 \sin(\pi t) \) indicates the length of the semi-major axis along the \( x \)-axis.
• Similarly, the coefficient 2 in \( 2 \cos(\pi t) \) corresponds to the semi-minor axis along the \( y \)-axis.

By evaluating the parametric equations, you can visualize how the shape twists around the origin, ultimately forming an oval shape. The elliptical path is periodic, completing a full round when \( t \) cycles from 0 to 2.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that describe rotational and oscillatory motion.

For the function \( \vec{r}(t) = \langle 3 \sin(\pi t), 2 \cos(\pi t) \rangle \), both functions guide the trajectory of the parametric curve. Here's why these functions are key:

• The \( \sin(\pi t) \) component affects movement along the \( x \)-axis. Its periodic nature cycles through values from -3 to 3.
• The \( \cos(\pi t) \) component dictates the \( y \)-axis movement, repeating its cycle from -2 to 2.

The interplay between these functions produces the elliptical path around the origin. Understanding this interaction helps demystify how sine and cosine contribute to the geometry of parametric curves.

Calculating Key Points

Calculating key points involves evaluating the vector function at specific values of \( t \). This helps us understand important characteristics of the curve.

For the ellipse prescribed by \( \vec{r}(t)=\langle 3 \sin (\pi t), 2 \cos (\pi t)\rangle \), compute vital positions:

• At \( t = 0 \), \( \vec{r}(0) = \langle 0, 2 \rangle \).
• At \( t = 1 \), \( \vec{r}(1) = \langle 0, -2 \rangle \).
• At \( t = 2 \), \( \vec{r}(2) = \langle 0, 2 \rangle \).

These computations give us points such as the starting position, the point furthest below the center, and the conclusion of one full loop. By connecting these points and considering the periodic nature of the functions, the entire path is revealed, simplifying the curve's sketch.