This document discusses vector-valued functions and some key concepts related to them. It provides examples of parametric curves and vector functions, how to sketch their graphs, and how to find derivatives and integrals of vector functions. It also introduces concepts like the unit tangent vector, unit normal vector, binormal vector, curvature, and radius of curvature for curves defined by vector functions.

1. VECTOR-VALUED FUNCTION
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Prepared by :MISS RAHIMAH JUSOH @
AWANG

2. ectorposition v
aasexpressedbecanD-3inequationcurveorlineA
kjir zyx 
:equationparametricin the
)(tfx  )(tgy 
)(thz 
kjir )()()((t) thtgtf 
)(),(),((t) thtgtfr

3. DOMAIN
Example 1
   
Determine the domain of the fo11owing function
cos ,1n 4 , 1
So1ution:
The first component is defined for a11 's.
The second component is on1y defined for 4.
The third component is on1y defined for
t t t t
t
t
  

r
 
1.
Putting a11 of these together gives the fo11owing domain.
1,4
This is the 1argest possib1e interva1 for which a11 three
components are defined.
t  


4. rahimahj@ump.edu.my
kjir )4(3)(ofgraphSketch the(b)
line.the
sketchThen(2,3,-1)?and(1,2,2)pointsthepasses
thatlinestraightaofequationlinetheisWhat(a)
2
ttt 
Example 2

5. kjir )23()2()1()(
232)21(
22)23(
11)12(Hence,
Then.(2,3,-1))z,(,1whenand
)2,2,1(),(,0nthat wheSuppose(a)
111
0,00






tttt
ttz
tty
ttx
,yxt
zyxt
rahimahj@ump.edu.my
Solution :
001 )( PtPPP 

6. rahimahj@ump.edu.my
Thus, the line:

7. rahimahj@ump.edu.my
3plane
on the4parabolatheisgraphtheis,chwhi
4z,3
thatfindweThus,
4,3,
arecurvetheofequationsParametric(b)
2
2
2




y
xz
xy
tzytx
Solution :

8. rahimahj@ump.edu.my

9. rahimahj@ump.edu.my
functionfollowingtheofeachofgraphSketch the
1,)( tt r(a) (b) ttt sin3,cos6)( r
Solution :
The first thing that we need to do is plug in a few values
of t and get some position vectors. Here are a few,
(a)
Example 3

10. rahimahj@ump.edu.my
The sketch of the curve is given as follows (red line).

11. rahimahj@ump.edu.my
Solution :
(b)
As in Question (a), we plug in some values of t.

12. rahimahj@ump.edu.my
The sketch of the curve is given as follows (red line).

13. functionfollowingtheofeachofgraphSketch the
kjir cttatat  sincos)(
Example 4
CIRCULAR HELIX

15. functionsscalaraisResult)()())((
functionsvectorareResults
)()())((
)()())((
)()())((
)()())((
Then
.offunctionscalaraisandoffucntionsareGandFSuppose
theorem.followingthehaveweThus
vectors.ofpropertiesloperationaeinherit thfunctionsVector
ttt
ttt
ttt
ttt
ttt
tt
GFGF
GFGF
FF
GFGF
GFGF














rahimahj@ump.edu.my
THEOREM 2.1

16. kji
kjikji
GFGF
GFGF
FGF
kjiG
kjiFGF
)sin5()
1
()(
5
1
)sin(
)()())(((i)
))(((iv)))(((iii)
))(((ii)))(((i)
find,5
1
)(and
sin)(bydefinedandfunctionsvectorFor the
2
2
2
t
t
ttt
t
tttt
ttt
tt
tet
t
tt
tttt
t







rahimahj@ump.edu.my
Example 4
Solution :

17. )()sin5()
sin
5(
5
1
sin
)5
1
()sin(
)()())(((iii)
)sin()()(
)())(()ii(
22
2
2
2
kji
kji
kjikji
GFGF
kji
FF
ttttt
t
t
t
t
t
ttt
t
tttt
ttt
teteet
tete
ttt
tt






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Solution :

18. rahimahj@ump.edu.my
Solution :
tt
t
tttt
ttt
sin51
)5
1
()sin(
)()())(((iv)
3
2



kjikji
GFGF

19. rahimahj@ump.edu.my
Example 5
246)(
1226)(
4)52(4)(3)(
if),(and),(Find
2
32
kiF
kjiF
kjiF
FF
tt
ttt
tt-tt
tt




Solution :

20. G
FG
FGF
G
FG
FGF
GF
GF
F
F
GF




dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
cc
dt
d
)()(iv
)(iii)(
)()ii(
)((i)
then,scalaraisc
andfunctionvectorabledifferentiareandIf:4.3Theorem
rahimahj@ump.edu.my
THEOREM 2.2

21. tttt
tttt
ttttt
dt
d
dt
d
dt
d
sin11cos)1(5
)cossin()310(
)sin(cos)5(
)()i(
2
2
32




jikji
jikji
G
FG
FGF
rahimahj@ump.edu.my
Example 6
)((iii)),((ii)),((i)
find,cossin)(,5)(If 32
FFGFGF
jiGkjiF


dt
d
dt
d
dt
d
ttttttt
Solution :

22. 53
2
2323
232
62100
2)()iii(
)cos11sinsin(5
t)sin3cos(-t)cos3sin(
0cossin
3110
0sincos
5
)()ii(
ttt
dt
d
dt
d
ttttt
tttttt
tt
tt
tt
ttt
dt
d
dt
d
dt
d






F
FFF
k
ji
kjikji
G
FG
FGF
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Solution :

23. rahimahj@ump.edu.my
INTEGRATION OF VECTOR
FUNCTIONS
))(())(())(()(
thenb],[a,onofand,,
functionsintegrablesomefrom)()()()(If
ise.componentwdonealsoisfunctionsvectorofnIntegratio
b
a
kjiF
kjiF
  

b
a
b
a
b
a
dtthdttgdttfdtt
thgf
thtgtft

24. rahimahj@ump.edu.my
Example 7
Solution :
kji
kji
kjiF
kjiF
F
802-42
])5()2[(
4)52()43()(
4t)52()43()(
if)(Find
3
1
4223
3
1
3
3
1
3
1
3
1
2
32
3
1




























 

ttttt
dttdttdtttdtt
tttt
dtt

25. rahimahj@ump.edu.my
 












b
a
dt
dy
dt
dx
L
22
 


















b
a
dt
dz
dt
dy
dt
dx
L
222
In 2-space
In 3-space
In general,
 
b
a
b
a
tL
dt
d
L )('or r
r

26. Notes: Smooth Curve
The graph of the vector function defined by
r(t) is smooth on any interval of t where is
continuous and .
The graph is piecewise smooth on an interval
that can be subdivided into a finite number of
subintervals on which r is smooth.
r
  0t r

27. rahimahj@ump.edu.my
Find the arc length of the parametric curve
4
3
0;2,sin,cos)( 33 
 tztytxa
10;2,,)(  
ttzeyexb tt
Find the arc length of the graph of r(t)
42;6
2
1
)()( 23
 ttttta kjir
20;2sin3cos3)()(  tttttb kjir
4
3
: LAns
1
: 
 eeLAns
58: LAns
132: LAns
Example 8
Example 9

29. If r(t) is a vector function that defines a smooth graph,
then at each point a unit tangent vector is
 
 
 
t
t
t



r
T
r
UNIT TANGENT VECTOR
   3
a) Find the derivative of 1 sin 2
b) Find the unit tangent vector at the point where 0.
t
t t te t
t

   

r i j k
Example 10

30. curve.thetont vectorunit tangetheiswhere
asbydenoted),(curvethevector to
normalunitprinciplethedefinewe,0If
T
Nr
T
t
dtd 
rahimahj@ump.edu.my
)('
)('
T
T
t
t
dtd
dtd
T
T
N 
UNIT NORMAL VECTOR

31. ).(curvethetoly,respectivevector
unitprincipaltheandnt vectorunit tangetheareandwhere
asdefinediscurveaofvectorbinormalThe
tr
NT
B
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BINORMAL VECTOR
NTB 

32.  
Find the unit normal and binormal
vectors for the circular helix
cos sint t t t  r i j k
Example 11

33. curve.thetont vectorunit tangetheiswhere
asdefinedis)(curvesmoothaofcurvatureThe
T
r t
rahimahj@ump.edu.my
)('
)('
t
t
dtd
dtd
r
T
r
T

CURVATURE
3

 


r r
r

34. Curvature is the measure of how “sharply” a curve r(t) in
2-space or 3-space “bends”.

35. Find the curvature of the helix traced out by
  2sin ,2cos ,4t t t tr
Example 12

36. Radius of Curvature
asdefinediscurvatureofradiusitsthen
),(curvesmooththeofcurvaturetheisIf

 tr


1


37. thentime,iswhere),r(ectorposition v
bygivencurvethealongmovesparticleaIf
tt
rahimahj@ump.edu.my
dt
d
t
r
v  )(velocity
2
2
)(onaccelerati
dt
d
dt
d
t
rv
a 
dt
ds
t  )(speed v

38. rahimahj@ump.edu.my
Example 13
.2when
particletheofonacceleratiandspeedvelocity,theFind
sincos)(
bygivenisafter timeparticleaofectorposition vThe
3


t
tttt
t
jir
Solution :
kji
kjiv
kji
r
v
1242.09.0
)2(3)2(cos)2(sin
2when
)3()(cos)(sin
velocityobtain thewe,w.r.tatingDifferenti
2
2




t
ttt
dt
d
t

39. kji
kjia
kji
v
a
a
v
129.00.42
)2(6)2sin()2cos(,2when
6)sin()cos(
bygivenisonacceleratiThe
04.12)2(91,2when
91)3()(cos)sin(
bygivenisany timeforspeedThe
4
42222






t
ttt
dt
d
t
tttt-
t



rahimahj@ump.edu.my

40. rahimahj@ump.edu.my
Example 14
kjir
r
kjiv
v


2)0(particle
theof)(ectorposition vtheFind
2cos)(
bygivenismotioninparticleaofVelocity
2
t
ttet t

41. rahimahj@ump.edu.my
     
C
Ce
cc
C
t
te
c
t
ctce
tdtdttdtet
dtd
t
t
t








i
kjir
kji
kji
kji
kjir
rv
2
)0(2sin
)0(
3
1
)0(
cCwhere
2
2sin
3
1
)
2
2sin
()
3
1
()(
2cos)(
havewe,Since
30
321
3
32
3
1
2
Solution :

42. kji
kjikjir
kji
kjii
r
)1
2
2sin
()1
3
1
()1(
)
2
2sin
(
3
1
)(
obtainweHence
C
2
obtainwe),0(ofegiven valutheusingBy
3
3




t
te
t
tet
C
t
t
rahimahj@ump.edu.my

43. Find the position vector R(t), given the
velocity V(t) and the initial position R(0) for
   2 2
; 0 4t
t t e t     V i j k R i j k
Example 15

44. rahimahj@ump.edu.my
rahimahj@ump.edu.my
“All our dreams can come true, if we have the
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