Question

Create a vector-valued function whose graph matches the given description. An ellipse, centered at (0,0) with vertical major axis of length 10 and minor axis of length \(3,\) traced once counterclockwise on \([0,2 \pi]\)

Short answer

The vector-valued function is \( \mathbf{r}(t) = \langle 1.5 \sin(t), 5 \cos(t) \rangle \) for \( t \in [0, 2\pi] \).

Step-by-step solution

Determine the parameters of the ellipse

Identify the details provided for the ellipse: the vertical major axis indicates that the length of the semi-major axis is 5 (half of 10) and is aligned along the y-axis, whereas the length of the semi-minor axis is 1.5 (half of 3) and is aligned along the x-axis.

Formulate the parametric equations for the ellipse

The parametric equations for an ellipse centered at (0,0) with a semi-major axis along the y-axis and a semi-minor axis along the x-axis are given by:\[x(t) = a \sin(t)\quad \text{and} \quad y(t) = b \cos(t),\]where \(a\) is the length of the semi-minor axis and \(b\) is the length of the semi-major axis.

Substitute the semi-major and semi-minor values into the equations

Using \(a = 1.5\) (the semi-minor axis) and \(b = 5\) (the semi-major axis), we substitute these into the parametric equations of the ellipse to get:\[x(t) = 1.5 \sin(t) \quad \text{and} \quad y(t) = 5 \cos(t).\]

Key concepts

Ellipse Properties

An ellipse is a fascinating shape in geometry that resembles a stretched out circle. The major characteristic of an ellipse are its axes. An ellipse has two axes: the major axis and the minor axis. The longer one is known as the major axis, and the shorter one is the minor axis.
For an ellipse centered at the origin, the equations take the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) represent the semi-minor and semi-major axes, respectively.
This implies:

• The length of the semi-major axis \(b\) is always greater.
• The length of each semi-axis is half the length of the full axis.
• The orientation of the major axis determines the alignment of the ellipse – whether vertically or horizontally aligned.

The unique property of ellipses where the sum of distances to two focal points is constant makes them widely useful in physical sciences, like astronomy, where they describe planetary orbits.

Vector-valued Functions

Vector-valued functions are functions that contain vectors as their output. When considering curves in a plane, vector-valued functions can describe the movement or shape of an object following a particular path.
A vector-valued function \(\mathbf{r}(t)\) can be represented as \[ \mathbf{r}(t) = \langle x(t), y(t) \rangle \],
where \(x(t)\) and \(y(t)\) are scalar functions of \(t\). Each scalar function describes the movement along their respective axis.
When used to outline the path of the ellipse, the vector function in parametric form becomes:

• \(x(t) = a \sin(t)\)
• \(y(t) = b \cos(t)\)

This formulation provides a dynamic way to describe the motion around the ellipse, as \(t\) varies from \(0\) to \(2\pi\). It effectively links geometry with algebra by seamlessly translating the properties of an ellipse into mathematical expressions.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures through the use of a coordinate system. It makes it easier to deal with curves and lines by expressing them in terms of equations.

When dealing with an ellipse, coordinate geometry allows us to describe its shape and position in the plane using equations and coordinates. By using equations such as \(x = a \sin(t)\) and \(y = b \cos(t)\), we can visualize an ellipse on a coordinate plane in a comprehensible way.

Coordinate geometry:

• Simplifies the expression and manipulation of shapes.
• Allows precise determination of distances and midpoints using formulas.
• Enables solving real-world problems that involve locating points and shapes in space.

In summary, coordinate geometry is a crucial tool in mathematics that bridges algebraic and geometric concepts to allow complex shapes like ellipses to be understood and used in calculations.