Question

Create a vector-valued function whose graph matches the given description. A circle of radius \(3,\) centered at \((5,5),\) traced clockwise once on \([0,2 \pi]\).

Short answer

The vector-valued function is \(\mathbf{r}(t) = \langle 5 + 3 \cos(t), 5 - 3 \sin(t) \rangle\) for \(t \in [0, 2\pi]\).

Step-by-step solution

Understanding the Circle Equation

A circle in the plane with radius \(r\) centered at point \((h, k)\) is described by the parametric equations: \(x = h + r \cos(\theta)\) and \(y = k + r \sin(\theta)\). The parameter \(\theta\) typically ranges from \(0\) to \(2\pi\) for a full circle transform.

Initial Vector-Valued Function

Given that the circle's radius is \(3\) and its center is \((5, 5)\), using the standard parametric equation, we have the vector-valued function \(\mathbf{r}(t) = \langle 5 + 3 \cos(t), 5 + 3 \sin(t) \rangle\). However, this describes counterclockwise motion.

Adjusting for Clockwise Motion

To trace the circle clockwise, we need to reverse the direction of tracing. We modify the angle in the sine component, i.e., \(\cos(t)\) remains as it is, but \(\sin(-t) = -\sin(t)\). Therefore, the vector-valued function becomes \(\mathbf{r}(t) = \langle 5 + 3 \cos(t), 5 - 3 \sin(t) \rangle\).

Verifying the Function

The resulting function \(\mathbf{r}(t) = \langle 5 + 3 \cos(t), 5 - 3 \sin(t) \rangle\) traces a complete circle of radius 3, centered at \((5,5)\) as parameter \(t\) varies from \(0\) to \(2\pi\), in the clockwise direction.

Key concepts

Parametric Equations

Parametric equations allow us to express coordinates - like \(x\) and \(y\) - in terms of a third variable, often denoted as \(t\) or \(\theta\). This method is particularly useful for describing paths and curves, as it can easily incorporate more complex movement and shape patterns beyond simple linear relationships.
For circles, parametric equations use trigonometric functions to relate the circle's radius and center to points along its circumference. The basic formula for a circle centered at \((h, k)\) with radius \(r\) is:

• \(x = h + r \cos(\theta)\)
• \(y = k + r \sin(\theta)\)

Here, \(\theta\) usually ranges from 0 to \(2\pi\), tracing out the full circle as it accounts for all possible angles around the circle's center. In terms of vector-valued functions, which combine these coordinates into a single function, we can easily visualize the path that a point will take over time.

Circle Equations

The equation of a circle helps describe every point on the circle's perimeter in relation to its center and radius. Geometrically, a circle is defined as the set of all points in a plane that are at a constant distance (the radius) from a fixed point (the center).
The standard form of the circle's equation in Cartesian coordinates is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
But when using parametric equations, we rely on trigonometric transformations to guide the points around the circle:

• This way, we can define the movement of a particle along the circle with these equations.
• It provides a way to animate or sketch the circle path efficiently in terms of time or angle changes.

The parametric version of the circle equations thus simplifies many geometrical challenges, making understanding and visualizing circular paths straightforward.

Trigonometric Transformations

Trigonometric transformations are crucial for modeling periodic functions like circles. By transforming trigonometric functions, we can manipulate how shapes are plotted using parametric equations.
A key aspect when dealing with circles and angles is deciding the direction of traversal. For instance, tracing a circle clockwise is a transformation of the standard counterclockwise movement. This involves altering the sign in the parametric equation for sine. Usually, reversing the direction can be done by negating the sine part, switching from \(\sin(t)\) to \(-\sin(t)\).

• This adjustment affects the path direction without changing the basic shape.
• It shows how effectively parametric equations and transformations can be adapted to fit specific requirements.

These transformations allow for depicting directionality as well as maintaining precision in defining intricate functions and their paths.