1

Chapter 13 – Vector Functions13.1 Vector Functions and Space Curves

13.1 Vector Functions and Space Curves

Objectives: Use vector -valued

functions to describe the motion of objects through space

Draw vector functions and their corresponding space curves

© jdannels - http://josephdannels.com/

13.1 Vector Functions and Space Curves

2

Vector FunctionsThe functions that we have been

using so far have been real-valued functions.We study functions whose values are vectors

because such functions are needed to describe curves and surfaces in space.

We will also use vector-valued functions to describe the motion of objects through space. ◦ In particular, we will use them to derive

Kepler’s laws of planetary motion.

13.1 Vector Functions and Space Curves

3

Definition – Vector functionA vector-valued function, or vector

function, is simply a function whose:

◦ Domain is a set of real numbers.

◦ Range is a set of vectors.

13.1 Vector Functions and Space Curves

4

Example 1 – pg. 845 #2Find the domain of the vector

function.

22( ) sin ln(9 )

2

tt t t

t

r i j k

13.1 Vector Functions and Space Curves

5

Definition – Component Functions If f(t), g(t), and h(t) are the components of the

vector r(t), then f, g, and h are real-valued functions called the component functions of r.

We can write: r(t) = ‹f(t), g(t), h(t)› = f(t) i + g(t) j + h(t) k

Note: We usually use the letter t to denote the independent variable because it represents time in most applications of vector functions.

13.1 Vector Functions and Space Curves

6

Definition – Limit of a VectorThe limit of a vector function r is

defined by taking the limits of its component functions as follows.

13.1 Vector Functions and Space Curves

7

Definition - ContinuousA vector function r is continuous at a if:

◦ In view of Definition 1, we see that r is continuous at a if and only if its component functions f, g, and h are continuous at a.

lim ( ) ( )t a

t a

r r

13.1 Vector Functions and Space Curves

8

Example 2Find the limit

0

1 1 1 3lim , ,

1

t

t

e t

t t t

13.1 Vector Functions and Space Curves

9

Space Curve Then, the set C of all points (x, y ,z) in space, where

x = f(t) y = g(t) z = h(t)

and t varies throughout the interval I is called a space curve.

The equations are called parametric equations of C.

t is called a parameter.

13.1 Vector Functions and Space Curves

10

VisualizationVector Functions and Space

CurvesThe Twisted Cubic CurveVisualizing Space Curves

13.1 Vector Functions and Space Curves

11

Example 3Find a vector equation and

parametric equations for the line segment that joins P to Q.

P (-2, 4, 0)

Q (6, -1, 2)

13.1 Vector Functions and Space Curves

12

Example 4 – pg. 846 # 29At what points does the curve

intersect the paraboloid ?

2( ) (2 )t t t t r i k2 2z x y

13.1 Vector Functions and Space Curves

13

Example 5 – pg 847 # 47If two objects travel through space along two

different curves, it’s often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of the two particles are given by the vector functions

for t0. Do the particles collide?

2 2 21 2( ) ,7 12, ( ) 4 3, ,5 6t t t t t t t t r r

13.1 Vector Functions and Space Curves

14

More Examples

The video examples below are from section 13.1 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 3◦Example 6

13.1 Vector Functions and Space Curves

15

Demonstrations

Feel free to explore these demonstrations below.

Four Space CurvesEquation of a Line in Vector Form

2D