13.2 13.1: Vector–Valued Functions

Consider a particle moving in R3 whose coordinates at time t are (x(t), y(t), z(t)). It is convenient to represent the particle’s path by the vector-valued function

Functions f(x) (with real number values) are often called scalar-valued to distinguish them from vector-valued functions.

Think of r(t) as a moving vector that points from the origin to the position of the particle at time t (Figure 1).

The parameter is often called t (for time), but we are free to use any other variable such as s or θ. It is best to avoid writing r(x) or r(y) to prevent confusion with the x- and y-components of r.

More generally, a vector-valued function is any function r(t) of the form in Eq. (1) whose domain is a set of real numbers and whose range is a set of position vectors. The variable t is called a parameter, and the functions x(t), y(t), z(t) are called the components or coordinate functions. We usually take as domain the set of all values of t for which r(t) is defined—that is, all values of t that belong to the domains of all three coordinate functions x(t), y(t), z(t). For example,

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The terminal point of a vector-valued function r(t) traces a path in R3 as t varies. We refer to r(t) either as a path or as a vector parametrization of a path. We shall assume throughout this chapter that the components of r(t) have continuous derivatives.

We have already studied special cases of vector parametrizations. In Chapter 12, we described lines in R3 using vector parametrizations. Recall that

r(t) = 〈x0, y0, z0〉 + tv = 〈x0 + ta, y0 + tb, z0 + tc〉

parametrizes the line through P = (x0, y0, z0) in the direction of the vector v = 〈a, b, c〉.

In Chapter 11, we studied parametrized curves in the plane R2 in the form

c(t) = (x(t), y(t))

Such a curve is described equally well by the vector-valued function r(t) = 〈x(t), y(t)〉. The difference lies only in whether we visualize the path as traced by a “moving point” c(t) or a “moving vector” r(t). The vector form is used in this chapter because it leads most naturally to the definition of vector-valued derivatives.

It is important to distinguish between the path parametrized by r(t) and the underlying curve traced by r(t). The curve is the set of all points (x(t), y(t), z(t)) as t ranges over the domain of r(t). The path is a particular way of traversing the curve; it may traverse the curve several times, reverse direction, or move back and forth, etc.

EXAMPLE 1: The Path versus the Curve

Describe the path

How are the path and the curve traced by r(t) different?

Solution As t varies from −∞ to ∞, the endpoint of the vector r(t) moves around a unit circle at height z = 1 infinitely many times in the counterclockwise direction when viewed from above (Figure 2). The underlying curve traced by r(t) is the circle itself.

Plot of r(t) = 〈cos t, sin t, 1〉.

A curve in R3 is also referred to as a space curve (as opposed to a curve in R2, which is called a plane curve). Space curves can be quite complicated and difficult to sketch by hand. The most effective way to visualize a space curve is to plot it from different viewpoints using a computer (Figure 3). As an aid to visualization, we plot a “thickened” curve as in Figures 3 and 5, but keep in mind that space curves are one-dimensional and have no thickness.

The curve r(t) = 〈t sin 2t cos t, t sin2 t, t cos t〉 for 0 ≤ t ≤ 4π, seen from three different viewpoints.

The projections onto the coordinate planes are another aid in visualizing space curves. The projection of a path r(t) = 〈x(t), y(t), z(t)〉 onto the xy-plane is the path p(t) = 〈x(t), y(t), 0〉 (Figure 4). Similarly, the projections onto the yz- and xz-planes are the paths 〈0, y(t), z(t)〉 and 〈x(t), 0, z(t)〉, respectively.

Projections of the helix r(t) = 〈− sin t, cos t, t〉.

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EXAMPLE 2: Helix

Describe the curve traced by r(t) = 〈−sin t, cos t, t〉 for t ≥ 0 in terms of its projections onto the coordinate planes.

Solution The projections are as follows (Figure 4):

  • xy-plane (set z = 0): the path p(t) = 〈− sin t, cos t, 0〉, which describes a point moving counterclockwise around the unit circle starting at p(0) = (0, 1, 0).

  • xz-plane (set y = 0): the path 〈− sin t, 0,t〉, which is a wave in the z-direction.

  • yz-plane (set x = 0): the path 〈0, cos t, t〉, which is a wave in the z-direction.

The function r(t) describes a point moving above the unit circle in the xy-plane while its height z = t increases linearly, resulting in the helix of Figure 4.

Every curve can be parametrized in infinitely many ways (because there are infinitely many ways that a point can traverse a curve as a function of time). The next example describes two very different parametrizations of the same curve.

EXAMPLE 3: Parametrizing the Intersection of Surfaces

Parametrize the curve obtained as the intersection of the surfaces x2 − y2 = z − 1 and x2 + y2 = 4 (Figure 5).

Intersection of surfaces x2 − y2 = z − 1 and x2 + y2 = 4.

Solution We have to express the coordinates (x, y, z) of a point on the curve as functions of a parameter t. Here are two ways of doing this.

First method: Solve the given equations for y and z in terms of x. First, solve for y:

The equation x2 − y2 = z − 1 can be written z = x2 − y2 + 1. Thus, we can substitute y2 = 4 − x2 to solve for z:

z = x2 − y2 + 1 = x2 − (4 − x2) + 1 = 2x2 − 3

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Now use t = x as the parameter. Then , z = 2t2 − 3. The two signs of the square root correspond to the two halves of the curve where y > 0 and y < 0, as shown in Figure 6. Therefore, we need two vector-valued functions to parametrize the entire curve:

Two halves of the curve of intersection in Example 3.

Second method: Note that x2 + y2 = 4 has a trigonometric parametrization: x = 2 cos t, y = 2 sin t for 0 ≤ t < 2π. The equation x2 − y2 = z − 1 gives us

z = x2 − y2 + 1 = 4 cos2 t − 4 sin2 t + 1 = 4 cos 2t + 1

Thus, we may parametrize the entire curve by a single vector-valued function:

EXAMPLE 4

Parametrize the circle of radius 3 with center P = (2, 6, 8) located in a plane:

  • Parallel to the xy-plane

  • Parallel to the xz-plane

Solution

  • Acircle of radius R in the xy-plane centered at the origin has parametrization 〈R cos t, R sin t〉. To place the circle in a three-dimensional coordinate system, we use the parametrization 〈R cos t, R sin t, 0〉.

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    Thus, the circle of radius 3 centered at (0, 0, 0) has parametrization 〈3 cos t, 3 sin t, 0〉. To move this circle in a parallel fashion so that its center lies at P = (2, 6, 8), we translate by the vector 〈2, 6, 8〉:

    r1(t) = 〈2, 6, 8〉 + 〈3 cos t, 3 sin t, 0〉 = 〈2 + 3 cos t, 6 + 3 sin t, 8〉

  • The parametrization 〈3 cos t, 0, 3 sin t〉 gives us a circle of radius 3 centered at the origin in the xz-plane. To move the circle in a parallel fashion so that its center lies at (2, 6, 8), we translate by the vector 〈2, 6, 8〉:

    r2(t) = 〈2, 6, 8〉 + 〈3 cos t, 0, 3 sin t〉 = 〈2 + 3 cos t, 6, 8 + 3 sin t〉

    These two circles are shown in Figure 7.

    Horizontal and vertical circles of radius 3 and center P = (2, 6, 8) obtained by translation.