Question

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is a circle (in a plane)? b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is an ellipse (in a plane)?

Short answer

Answer: a. For the path to be a circle: a = h - R, b = k, c = k - R, d = h, e = 0, and f = constant. b. For the path to be an ellipse: a = h - A, b = k, c = k - B, d = h, e = 0, and f = constant.

Step-by-step solution

Write down the parametric equation for the given path

The object's path is given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\)

Analyze the conditions for a plane

In order for the path to lie in a plane, all points on the path must have the same z-coordinate. To make the z-component in the path equation a constant, we must have \(e = 0\) and \(f = \text{constant}\).

Write parametric equations for a circle and ellipse

A circle of radius R with center (h, k) can be described using the parametric equation: \((x-h)=R\cos t\) and \((y-k)=R\sin t\). An ellipse with a semi-major axis of length A and semi-minor axis of length B, centered at (h, k), can be described using the parametric equation: \((x-h)=A\cos t\) and \((y-k)=B\sin t\).

Analyze the conditions for a circle

For the given path to be a circle in a plane, it should be of the form \((x-h)=R\cos t\) and \((y-k)=R\sin t\). By comparing the given path with the path of a circle, we get the conditions: a = h - R, b = k, c = k - R, and d = h.

Analyze the conditions for an ellipse

For the given path to be an ellipse in a plane, it should be of the form \((x-h)=A\cos t\) and \((y-k)=B\sin t\). By comparing the given path with the path of an ellipse, we get the conditions: a = h - A, b = k, c = k - B, and d = h. a. So, the conditions on a, b, c, d, e, and f to guarantee that the path is a circle are: a = h - R, b = k, c = k - R, d = h, e = 0, and f = constant. b. The conditions on a, b, c, d, e, and f to guarantee the given path is an ellipse are: a = h - A, b = k, c = k - B, d = h, e = 0, and f = constant.

Key concepts

Circle Conditions

When we talk about a circle, we envision a perfectly round shape where every point on its edge is equidistant from a fixed center point. In mathematics, this is a basic 2D geometric shape. In parametric form, a circle is often represented using equations based on trigonometric functions.
For a path to describe a circle in a plane using vector functions, certain conditions must be satisfied.

• First, any changes in the z-coordinate must be eliminated to ensure the path lies flatly in a 2D plane. This is achieved by setting the coefficients associated with the z-components to zero or a constant value, specifically: \( e = 0 \) and \( f = \text{constant} \).
• Additionally, the coefficients of cosine and sine functions in the parametric equations must form a circle's pattern, meaning they should equal the radius of the circle, \( R \), when combined.
• Thus, the resulting conditions for a circle path are \( a = h - R, \, b = k, \, c = k - R, \, ext{and } d = h \, \), with z-coefficients suitably set.

In summary, these conditions aligned together ensure that the path traces a circular motion smoothly within the plane.

Ellipse Conditions

An ellipse is like an elongated circle. It's a 2D shape where the distances from any point on its perimeter to two distinct points (foci) add up to the same value. Ellipses have broader and narrower sections, determined by their axes, similar to stretching a rubber band around two thumbtacks.
For a vector function to describe an ellipse within a plane, it requires specific conditions. These conditions transform the circle's parametric form into the one needed for an ellipse.

• To maintain a 2D plane and eliminate vertical displacement, the z-components' coefficients again need to be constants, specifically \( e = 0 \) and \( f = \text{constant} \).
• The path coefficients must reflect the ellipse's semi-major and semi-minor axes, \ A \ and \ B \, respectively. These act as stretched radius values along their respective axes.
• Thus, the conditions become \( a = h - A, \, b = k, \, c = k - B, \, ext{and } d = h \, \), which together define how the ellipse is oriented in the coordinate plane.

The combination of these conditions guarantees the path will map out an elliptical shape.

Vector Functions

A vector function is a function that assigns a vector to each point in its domain. In the context of parametric equations, it describes how an object moves in space by linking each point in time to a specific vector. For a path given by \( \mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\), the vector function provides a way to explore complex paths that objects trace over time.
The uses of vector functions in mathematics and physics are vast:

• They allow the representation of curves and paths in multidimensional spaces.
• They provide a detailed understanding of motion, particularly in physics, through analysis of changes in vector components over time.

When working with paths like circles and ellipses, vector functions utilize parametric equations for simplified expressions of the paths' geometry and computation.
Understanding vector functions equips us with tools to explore and manipulate motion and shapes more effectively, capturing both their direction and magnitude dynamically as time progresses.