Question

Represent the plane curve by a vectorvalued function. (There are many correct answers.) $$ \frac{x^{2}}{16}+\frac{y^{2}}{9}=1 $$

Short answer

The vector-valued function that represents the given plane curve is: \(\vec{r}(t) = <4cos(t), 3sin(t)>\). Note that t here is the parameter which will range between 0 and 2Pi for a complete traverse around the ellipse.

Step-by-step solution

Identify the Type of Curve

The equation given is for an ellipse. An ellipse is a curve that is the locus of all points in the plane the sum of whose distances from two fixed points (called the foci) is a constant. The given ellipse equation is of the standard form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where a and b are the lengths of the semi-major and semi-minor axes respectively.

Establish parametric equations for x and y

By using the identity \(\sin^2(t) + \cos^2(t) = 1\), parametric equations can be derived for an ellipse equation. That is, let \(x = a\cos(t)\) and \(y = b\sin(t)\), where t is the parameter.

Substitute for a and b

We know from the given equation that the semi-major axis length a is 4, and the semi-minor b is 3, substituting these values, we get: \(x = 4\cos(t)\) and \(y = 3\sin(t)\).

Convert into Vector Form

The final step is to convert these parametric equation representations into vector form. In 2-dimensional space, the vector is a 2-component vector as \(\vec{r}(t) = \), substituting x(t) and y(t) values, we get: \(\vec{r}(t) = <4cos(t), 3sin(t)>\)

Key concepts

Ellipse Equation

An ellipse is a geometric shape that looks like a stretched circle. It can be thought of as the path traced by a point moving in such a way that the sum of its distances from two fixed points, called foci, is constant. Imagine taking a loop of string, pinning it at two points on a sheet of paper and stretching it with a pencil to draw a curve; that curve is an ellipse.

When we want to describe an ellipse mathematically in the Cartesian plane, we use the ellipse equation, which has the standard form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). Here, \( a \) and \( b \) represent the lengths of the semi-major axis and the semi-minor axis respectively. The semi-major axis is the longest radius of the ellipse, and the semi-minor axis is the shortest. When you come across an ellipse equation, identifying \( a \) and \( b \) is crucial because they determine the shape and size of the ellipse.

Parametric Equations

Parametric equations are a way of describing a mathematical object using a parameter, which is especially useful for defining curves, such as the ellipse in our exercise. Instead of expressing a relationship directly between \( x \) and \( y \), we introduce a third variable, usually \( t \), as the parameter that represents another dimension, such as time.

By expressing \( x \) and \( y \) in terms of \( t \), we create a pair of equations known as parametric equations. For the ellipse with equation \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), we can use \( x = a\cos(t) \) and \( y = b\sin(t) \) as the parametric equations. These equations leverage the property that \( \cos^{2}(t) + \sin^{2}(t) = 1 \) for all values of \( t \), making them particularly elegant for describing ellipses and other trigonometric curves.

Vector Representation of Curves

Vector representation is an alternate form to describe curves in mathematics and physics, providing a dimension-agnostic notation that works for planes and higher-dimensional spaces. Think of it as giving directions for moving along the curve in a coordinate space.

To transition from parametric equations to vector representation, we use vectors to encapsulate the coordinates at each point along the curve. For a two-dimensional curve, such as our ellipse, we describe the location of any point on the curve at a given parameter \( t \) as \( \vec{r}(t) \) where \( \vec{r}(t) = \). In the context of our exercise, the vector-valued function that represents the ellipse is \( \vec{r}(t) = <4\cos(t), 3\sin(t)> \), which tells us the exact position of a point on the ellipse at any given moment, parameterized by \( t \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within their domain. These identities are useful in various areas of mathematics, including solving complex equations and transforming expressions. One of the most fundamental and frequently used identities is \( \sin^{2}(t) + \cos^{2}(t) = 1 \), which is referred to as the Pythagorean identity.

Parametric equations for the ellipse capitalize on this identity to link \( x \) and \( y \) to the parameter \( t \) while ensuring the equation of the ellipse is satisfied. By choosing \( x(t) = a\cos(t) \) and \( y(t) = b\sin(t) \) as our parametric representations, we guarantee that for any value of \( t \) we choose, the point \( (x(t), y(t)) \) will indeed lie on the ellipse.