18 Vector Valued Functions.pdf

Vector-Valued Functions

Mathematics 54 - Elementary Analysis 2

Introduction to Vector-Valued Functions

Definition.

Avector-valued function~R, or simply avector function, is a function whose domain is a set of real numbers and whose range is a set of vectors.

Examples.

~R(t)=t2ˆı₊(t₊1) ˆ; ~S(t)= 〈cost, sint,t_〉

Remarks.

1 A functionf:R_→Ris called areal-valued functionor ascalar

function.

2 _Let~R₍t₎₌­x₍t_{) ,}y₍t_{) ,}z₍t₎®_{, where}x_,y_andz_{are real-valued functions.}

Thedomainof~R, denoted by dom~R, is given by

Domain of Vector Functions

Example

Determine the domain of the following vector functions:

1 ~_{R(t)=ln(t−1)ˆı+} 1 t2_−t−6ˆ

2 ~_S(t)= ¿ ₁

t+5,

p

4−t,et À

3 ~_A(t)= D

sin(t₋2),pt2_{−t, logt}E

Solution.

1 dom~R =(1,+∞)∩R\{−2, 3}

=(1, 3)∪(3,+∞)

2 dom~S =R\{−5}∩(−∞, 4]∩R =(−∞,−5)∪(−5, 4]

Graphing Vector Functions

Definition.

Thegraphof−→R(t)_{= 〈}x(t),y(t),z(t)_〉is the curve in space traced by the endpoints of the vector−→Rin its standard position, for allt∈dom→−R(t), in

the direction of the increasing parametert.

Remark

For the vector-valued function−→R(t)_{= 〈}x(t),y(t),z(t)_〉, the equations

Graphing Vector Functions

Consider

− →_R(t)

Graphing Vector Functions

Consider

~S(t)_{= 〈}cost, sint,t〉.(circularhelix)

Some points:

t =0,π₆,π₄,π₃;

t =π₂,2₃π,3₄π,5₆π;

t ₌π,7π₆,5π₄,4π₃;

Remark

The graph of the vector function~R(t)_{= 〈}f(t),g(t),h(t)_〉is identical to the graph of the parametric curve

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

Example

Sketch the graph of~R(t)₌sin(t)ˆı₊sin3(t) ˆ.

Example

Vector Equations

Consider the line in space passing throughP0(x0,y0,z0) and parallel to a

vector~v= 〈a,b,c〉

Vector Equations

Find a vector equation of the line segment that joins the pointsP(1,₋2, 3)

andQ(4, 2, 2). Take~v=−→PQ= 〈3, 4,−1〉.

line:~r(t)= 〈1+3t,−2+4t, 3−t〉

Vector Equations

Example

Find a vector equation for the curve of intersection of the cylinder

x2

9 +

y2

4 =1 and the planey+z=3.

Parametrization of the cylinder:

x₌3 cost, andy₌2 sint, since

cos2t+sin2t=x

2

9 +

y2

4 =1.

Thus,z=3−y=3−2 sint.

Hence, a vector equation for the curve is

− →_R(t)

Operations on Vector Functions

Let~Fand~Gbe vector functions andf be a real-valued function.

1 _Addition:(~F₊~G_{) (}t_{) :=}~F₍t₎₊~G₍t₎

2 _{Dot Product:(}~_F·~_{G) (t) :=}~_F(t)·~_G(t)

3 Cross Product:(~F_×~G) (t) :₌~F(t)_×~G(t)

4 _{Scalar Product:(f}~_{F) (t) :=f(t)}~_F(t)

Operations on Vector Functions

Example

Given−→F(t)=­

t+1,t2−1,t−1®

,−→G(t)= 〈t−1, 1,t+1〉, andf(t)=et−1.

(→−F₋→−G) (t) ₌­

t₊1,t2₋1,t₋1®

− 〈t₋1, 1,t₊1〉 =2ˆı+(t2−2) ˆ−2ˆk

(→−F×→−G) (t) = 〈t+1,t2−1,t−1〉 × 〈t−1, 1,t+1〉

=¡¡t2₋1¢

(t₊1)₋(t₋1)¢

ˆ

ı₋¡

(t₊1)2₋(t₋1)2¢

ˆ

 +¡

(t+1)−(t−1)¡

t2−1¢¢_ˆ k =¡

t3+t2−2t¢

ˆ

ı−(4t) ˆ+¡

−t3+t2+2t¢_ˆ k

(−→F _◦f) (t) =→−F¡ et₋1¢

= D

¡ et₋1¢

+1,¡ et₋1¢2

−1,¡ et₋1¢

−1E

Exercises

1 _{Find the domain of}~R₍t₎₌ 1−t t2₋₄ˆı−e

sint_ˆ.

2 _{Find the domain of}~_S(t)= ¿

ln(t+2),3

t, p

9−t2

À

.

3 _{Sketch the graph of}~_{R(t)= 〈4t}2_{, 3,t〉.}

4 _{Find the vector equation of the line segment passing through the}

points (1, 3,−2) and (3, 2, 4).

5 _{Find the vector equation of the line of intersection of the planes}

3x₋y₊z₌3 and 2x₊2y₋z₌5.

6 Find the parametric equations of the curve of intersection of the

paraboloidsz₌x2₊y2and 4₌3x2₊y2₊z.

7 _Let~_{F(t)= 〈sin 2t, cos 4t,t〉andg(t)=t−}π