91Ó°ÊÓ

Parametric equations are a way to describe a curve by using a set of equations that express the coordinates of the points on the curve as functions of a parameter. This approach allows for a more flexible definition of curves, especially when compared to the standard Cartesian form of using just x and y coordinates.
In the exercise, the vector function \( \mathbf{r}(t) = \langle t, 2t \rangle \) represents the parametric equations for the curve. Here, the parameter \( t \) is used to describe both coordinates of a point:

• \( x(t) = t \)
• \( y(t) = 2t \)

These equations simplify to show how each coordinate depends on the parameter \( t \), leading us to express the curve in the Cartesian coordinate system by substituting \( t \) as \( x \), giving \( y = 2x \).

The use of parametric equations is particularly useful for tracing the direction of the curve as the parameter changes. This is known as the curve's orientation.

Curve sketching involves plotting a graph to visually represent a set of equations, thus providing valuable insights into the behavior and nature of a curve. With parametric equations, it becomes essential to analyze how curves change as parameter \( t \) varies within a given interval.
In the given problem, the task was to sketch the curve described by the equations for \( 0 \leq t \leq 1 \). By translating the parametric form \( \mathbf{r}(t) = \langle t, 2t \rangle \), we derived the linear equation \( y = 2x \), which is a straightforward representation in Cartesian form.
To sketch this curve, we identify and plot key points based on the endpoints of the parameter interval, such as \((0, 0)\) and \((1, 2)\). Drawing a line segment between these points accurately depicts the section of the line defined by the parameter \( t \).
Always remember, sketching parametric curves accurately will provide a clear picture of the curve's orientation and define its exact segment in the Cartesian plane.

Cartesian coordinates are a familiar way to locate points on a plane using two perpendicular axes, usually labeled as \(x\) and \(y\). By converting parametric equations into Cartesian form, one can analyze curves intuitively, as it translates a curve into a well-known structure.In this exercise, after determining the parametric equations \( x(t) = t \) and \( y(t) = 2t \), the next step was to express these in Cartesian form. By substituting \( t = x \) into the equation for \( y \), we derive \( y = 2x \). This line equation clearly fits into the Cartesian framework, making it easier to understand and sketch.
Within the bounds of \(0 \leq t \leq 1\), we analyzed the line segment between the points \((0, 0)\) and \((1, 2)\). The transformation from parametric to Cartesian coordinates not only helps in sketching the curve but also clarifies the spatial relationship between elements in a visually intuitive form.
Understanding how to move between these two methods and representations can greatly enhance one's ability to interpret and solve problems related to vector functions and curves.