So far, we have seen several different examples of curves in space, including traces and contours of functions of two variables, as well as lines in 3-space. Recall that for a line through a fixed point in the direction of vector , we may express the line parametrically through the single vector equation
Like lines, other curves in space are one-dimensional objects, and thus we aspire to similarly express the coordinates of points on a given curve in terms of a single variable. Vectors are a perfect vehicle for doing so — we can use vectors based at the origin to identify points in space, and connect the terminal points of these vectors to draw a curve in space. This approach will allow us to draw an incredible variety of graphs in 2- and 3-space, as well as to identify and describe curves in -space for any . It will also allow us to represent traces and cross sections of surfaces in space.
Based on the pictures from parts (a) and (b), sketch the set of terminal points of all of the vectors of the form , where assumes values from 0 to . What is the resulting figure? Why?
Suppose we sketched the terminal points of all vectors of the form , where assumes values from 0 to . How does the resulting picture differ from the one in part (c)? What about for from 0 to ?
Consider the curve shown in Figure 9.6.1. As in Preview Activity 9.6.1, we can think of a point on this curve as resulting from a vector from the origin to the point. As the point travels along the curve, the vector changes in order to terminate at the desired point. A few still pictures of this motion are shown in Figure 9.6.1.
Thus, we can think of the curve as a collection of terminal points of vectors emanating from the origin. We therefore view a point traveling along this curve as a function of time , and define a function whose input is the variable and whose output is the vector from the origin to the point on the curve at time . In so doing, we have introduced a new type of function, one whose input is a scalar and whose output is a vector.
The terminal points of the vector outputs of then trace out the curve in space. From this perspective, the ,, and coordinates of the point are functions of time, , say
and
and thus we have three coordinate functions that enable us to represent the curve. The variable is called a parameter and the equations ,, and are called parametric equations (or a parameterization of the curve). The function whose output is the vector from the origin to a point on the curve is defined by
Note that the input of is the real-valued parameter and the corresponding output is vector . Such a function is called a vector-valued function because each real number input generates a vector output. More formally, we state the following definition.
A vector-valued function is a function whose input is a real parameter and whose output is a vector that depends on . The graph of a vector-valued function is the set of all terminal points of the output vectors with their initial points at the origin.
Note particularly that every set of parametric equations determines a vector-valued function of the form
and every vector-valued function defines a set of parametric equations for a curve. Moreover, we can consider vector-valued functions and parameterizations in ,, or indeed a real space of any dimension. As a reminder, in Section 9.5, we determined the parametric equations of a line in space using a point and a direction vector. For a nonlinear example, the curve in Figure 9.6.1 has the parametric equations
The same curve can be represented with different parameterizations. Use appropriate technology to plot the curves generated by the following vector-valued functions for values of from to . Compare and contrast the graphs — explain how they are alike and how they are different.
The examples in Activity 9.6.2 illustrate that a parameterization allows us to look not only at the graph, but at the direction and speed at which the graph is traversed as changes. In the different parameterizations of the circle, we see that we can start at different points and move around the circle in either direction. The calculus of vector-valued functions — which we will begin to investigate in Section 9.7 — will enable us to precisely quantify the direction, speed, and acceleration of a particle moving along a curve in space. As such, describing curves parametrically will allow us to not only indicate the curve itself, but also to describe how motion occurs along the curve.
Using parametric equations to define vector-valued functions in two dimensions is much more versatile than just defining as a function of . In fact, if is a function of , then we can parameterize the graph of by
and thus every single-variable function may be described parametrically. In addition, as we saw in Preview Activity 9.6.1 and Activity 9.6.2, we can use vector-valued functions to represent curves in the plane that do not define as a function of (or as a function of ). (As a side note: vector-valued functions make it easy to plot the inverse of a one-to-one function in two dimensions. To see how, if defines a one-to-one function, then we can parameterize this function by . Since the inverse function just reverses the role of input and output, a parameterization for is .)
Vector-valued functions can be used to generate many interesting curves. Graph each of the following using an appropriate technological tool, and then write one sentence for each function to describe the behavior of the resulting curve.
Recall from our earlier work that the traces and level curves of a function are themselves curves in space. Thus, we may determine parameterizations for them. For example, if , the trace of the function is given by setting and letting be parameterized by the variable ; then, the trace is the curve whose parameterization is
How do your responses change to all three of the preceding questions if you instead consider the function defined by ? (Hint for generating one of the parameterizations: .)
A vector-valued function is a function whose input is a real parameter and whose output is a vector that depends on . The graph of a vector-valued function is the set of all terminal points of the output vectors with their initial points at the origin.
Every vector-valued function provides a parameterization of a curve. In , a parameterization of a curve is a pair of equations and that describes the coordinates of a point on the curve in terms of a parameter . In , a parameterization of a curve is a set of three equations ,, and that describes the coordinates of a point on the curve in terms of a parameter .
Find a parametrization of the circle of radius in the xy-plane, centered at the origin, oriented clockwise. The point should correspond to . Use as the parameter for all of your answers.
Find a vector parametrization of the circle of radius in the xy-plane, centered at the origin, oriented clockwise so that the point corresponds to and the point corresponds to .
A bicycle wheel has radius R. Let P be a point on the spoke of a wheel at a distance d from the center of the wheel. The wheel begins to roll to the right along the the x-axis. The curve traced out by P is given by the following parametric equations:
A standard parameterization for the unit circle is , for .
Find a vector-valued function that describes a point traveling along the unit circle so that at time the point is at and travels clockwise along the circle as increases.
Find a vector-valued function that describes a point traveling along the unit circle so that at time the point is at and travels counter-clockwise along the circle as increases.
Find a vector-valued function that describes a point traveling along the unit circle so that at time the point is at and travels clockwise along the circle as increases.
Find a vector-valued function that describes a point traveling along the unit circle so that at time the point is at and makes one complete revolution around the circle in the counter-clockwise direction on the interval .
Let and be positive real numbers. You have probably seen the equation that generates an ellipse, centered at , with a horizontal axis of length and a vertical axis of length .
Explain why the vector function defined by , is one parameterization of the ellipse .
Use the traces and contours you’ve just investigated to create a wireframe plot of the surface generated by . In addition, write two sentences to describe the characteristics of the surface.
Determine the point of intersection of the lines given by
Then, find a vector-valued function that parameterizes the line that passes through the point of intersection you just found and is perpendicular to both of the given lines.