CHAPTER

11Vector-Valued Functions

11

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11.1 VECTOR-VALUED FUNCTIONS11.2 THE CALCULUS OF VECTOR-VALUED FUNCTIONS11.3 MOTION IN SPACE11.4 CURVATURE11.5 TANGENT AND NORMAL VECTORS11.6 PARAMETRIC SURFACES

11.1 VECTOR-VALUED FUNCTIONS

Preliminaries

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For the circuitous path ofthe airplane, it is convenient to describe the airplane’s location at any given time by the endpoint of a vector whose initial point is located at the origin (a position vector).Notice that a function that gives us a vector in V3 for each time t would do the job nicely. This is the concept of a vector-valued function.

DEFINITION

11.1 VECTOR-VALUED FUNCTIONS

1.1

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A vector-valued function r(t) is a mapping from its domain D to its range ⊂ R ⊂ V3, so that for each t in D, r(t) = v for exactly one vector v ∈ R.

We can always write a vector-valued function asr(t) = f (t)i + g(t)j + h(t)k, (1.1)

for some scalar functions f, g and h (called the component functions of r).

We can likewise define a vector-valued function r(t) in V2 by r(t) = f (t)i + g(t)j, for some scalar functions f and g.

11.1 VECTOR-VALUED FUNCTIONS

Preliminaries

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For each t, we regard r(t) as a position vector.

The endpoint of r(t) then can be viewed as tracing out a curve.

Observe that for r(t) as defined in (1.1), this curve is the same as that described by the parametric equations x = f (t), y = g(t) and z = h(t).

In three dimensions, such a curve is referred to as a space curve.

EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.1 Sketching the Curve Defined by a Vector-Valued Function

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Sketch a graph of the curve traced out by the endpoint of the two-dimensional vector-valued function

r(t) = (t + 1)i + (t2 − 2)j.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.1 Sketching the Curve Defined by a Vector-Valued Function

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Substituting some values for t, we have r(0) = i − 2j = 1,−2,r(2) = 3i + 2j = 3, 2 and r(−2) = −1, 2.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.1 Sketching the Curve Defined by a Vector-Valued Function

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The endpoints of all position vectors r(t) lie on the curve C, described parametrically by

Eliminate the parameter:

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.1 Sketching the Curve Defined by a Vector-Valued Function

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The small arrows marked on the graph indicate the orientation, that is, the direction of increasing values of t. If the curve describes the path of an object, then the orientation indicates the direction in which the object traverses the path.

EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.2 A Vector-Valued Function Defining an Ellipse

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Sketch a graph of the curve traced out by the endpoint of the vector-valued function

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.2 A Vector-Valued Function Defining an Ellipse

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EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.3 A Vector-Valued Function Defining an Elliptical Helix

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Plot the curve traced out by the vector-valued function r(t) = sin ti − 3 cos tj + 2tk, t ≥ 0.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.3 A Vector-Valued Function Defining an Elliptical Helix

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EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.4 A Vector-Valued Function Defining a Line

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Plot the curve traced out by the vector-valued function

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.4 A Vector-Valued Function Defining a Line

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Recognize these equations as parametric equations for the straight line parallel to the vector 2,−3,−4 and passing through the point (3, 5, 2).

EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.5 Matching a Vector-Valued Function to Its Graph

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Match each of the vector-valued functions

f1(t) = cos t, ln t, sin t,f2(t) = t cos t, t sin t, t, f3(t) = 3 sin 2t, t, t and f4(t) = 5 sin3 t, 5 cos3 t, t

with the corresponding computer-generated graph (on the following slide).

EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.5 Matching a Vector-Valued Function to Its Graph

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f1(t) = cos t, ln t, sin t,

f2(t) = t cos t, t sin t, t,

f3(t) = 3 sin 2t, t, t and

f4(t) = 5 sin3 t, 5 cos3 t, t

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.5 Matching a Vector-Valued Function to Its Graph

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f1(t) = cos t, ln t, sin t,

f2(t) = t cos t, t sin t, t,

f3(t) = 3 sin 2t, t, t and

f4(t) = 5 sin3 t, 5 cos3 t, t

f2

f1

f3

f4

11.1 VECTOR-VALUED FUNCTIONS

Arc Length in

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A natural question to ask about a curve is, “How long is it?”

Note that the plane curve traced out exactly once by the endpoint of the vector-valued function r(t) = f (t), g(t), for t [∈ a, b] is the same as the curve defined parametrically by x = f (t), y = g(t).

Recall from section 10.3 that if f, f' , g and g' are all continuous for t [∈ a, b], the arc length is given by

11.1 VECTOR-VALUED FUNCTIONS

Arc Length in

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Consider a space curve traced out by the endpoint of the vector-valued function r(t) = f (t), g(t), h(t), where f, f' , g, g', h and h' are all continuous for t [∈ a, b] and where the curve is traversed exactly once as t increases from a to b.

The arc length of the space curve is given by:

(See the text for the derivation of this result.)

EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.6 Computing Arc Length in

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Find the arc length of the curve traced out by the endpoint of the vector-valued function r(t) = 2t, ln t, t2, for 1 ≤ t ≤ e.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.6 Computing Arc Length in

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First, notice that for x(t) = 2t, y(t) = ln t and z(t) = t2, we have x'(t) = 2, y'(t) = 1/t and z'(t) = 2t, and the curve is traversed exactly once for 1 ≤ t ≤ e.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.6 Computing Arc Length in

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EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.7 Approximating Arc Length in

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Find the arc length of the curve traced out by the endpoint of the vector-valued function r(t) = e2t , sin t, t, for 0 ≤ t ≤ 2.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.7 Approximating Arc Length in

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First, note that for x(t) = e2t , y(t) = sin t and z(t) = t, we have x'(t) = 2e2t , y'(t) = cos t and z'(t) = 1, and that the curve is traversed exactly once for 0 ≤ t ≤ 2

Approximate the integral using Simpson’s Rule or the numerical integration routine built into your calculator:

s ≈ 53.8.

EXAMPLE

11.1 VECTOR-VALUED FUNCTIONS

1.8 Finding Parametric Equations for an Intersection of Surfaces

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Find the arc length of the portion of the curve determined by the intersection of the cone

and the plane y + z = 2 in the first octant.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.8 Finding Parametric Equations for an Intersection of Surfaces

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EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.8 Finding Parametric Equations for an Intersection of Surfaces

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The portion of the parabola in the first octant must have x ≥ 0 (so t ≥ 0), y ≥ 0 (so t2 ≤ 4) and z ≥ 0 (always true).

This occurs if 0 ≤ t ≤ 2.

EXAMPLE

Solution

11.1 VECTOR-VALUED FUNCTIONS

1.8 Finding Parametric Equations for an Intersection of Surfaces

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