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8/12/2019 MATH 37 UNIT 5.1 (1)
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UNIT 5
VECTORS, L INESand PLANES in
SPACE
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OBJECTIVES
By the end of the unit, you must be
able to: enumerate and apply properties of
vectors in the plane and in space;
perform and interpret vector
operations;
find the equations of a line and
equation of a plane in space; and
identify and sketch cylinders and
quadric surfaces.
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5.1
VECTORS IN 2Dand IN 3D
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NOTION
VECTOR
SCALAR
quant i ty that has both
magn i tude and d irect ion
quant i ty that on ly hasmagni tude
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Examples
SCALARS
MATH 37 GRADE
speedlength
t imetemperature
densi ty
mass
energy
SCALARS VECTORS
magnet ic f ield
veloci tydisplacement
accelerat ionforce
elec tr ic f ieldmomentum
LIFE
VECTORS
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L IFE IS NOT JUST
ABOUT MAGNITUDE.
IT NEEDS DIRECTION.
JUST FOR FUN
-someth ing I overheard
MAGNITUDEDIRECTION
LIFE
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Geometr ic representat ion
magni tude
direct ion
in i t ial po int
term inal po int
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Example.
Cons ider a vecto r w i th
ini t ial po int at and
term inal po int at .
A42 ,
65,Determ ine its magn i tude and
direct ion.
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Solut ion
1 2 3 4 5
1
2
34
5
-1-2-3-4-5-1
-2
-3
-4
-5
65,
42 ,
magni tude
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Solut ion (cont inued)
magn itude of A
6542 ,,,d
246225 2
102
7
149
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Solut ion (cont inued)
1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-542 ,
65,
10
7
tan 710
7
10tanArc
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Solut ion (cont inued)
Hence, vecto r has a
magn i tude of and inthe d irect ion o f .
A149
7
10tanArc
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Representations . . .
A vecto r has several representat ions
on the plane depend ing on the ini t ial
and term inal poin t .
Posi t ion Representat ion
in i t ial po int at the or ig in
direct ion is measu red from theposi t ive x-axis
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Example. The fol low ing are
d i f feren t representat ions o f one
vector.
In i t ial po int Term inal po int
23, 11 ,
15 , 41
,51, 25,
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1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
23,
11 ,
51,
25,
41 , 34 ,
15 ,
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Vecto r in the p lane
A vectoris an ordered pair of
real numbers . aand bare called as componentso fthe vec to r.
b,a
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Representat ion
Posi t ion representat ion o f
in i t ial poin t :
term inal po int :
b,a
00, b,a
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Example 1.
Determ ine the components o f
the vecto r w i th ini t ial po int at
and term inal po int at
.
42, 65,Solut ion:
itit yy,xx 4625 ,
107
,
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1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-542,
65,107,
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Example 2.
Determ ine the components o f
the vecto r w i th ini t ial po int at
and term inal po int at
.
23,
11 ,Solut ion:
itit yy,xx 2131 ,
34,
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1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
23
,
11 ,34,
34 ,
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Equali ty o f vecto rs
Two vecto rs are equal i f theirmagni tudes and d irect ions are
equal.
Vectors and are
equali f and on ly i f and.
b,a
ca d,c
db
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Magn i tude and direct ion
Consider vecto r A.
The magni tudeo f A, , is theleng th o f any of i ts
representat ions.
A
The d irect ion angle o f A, , is
the measu re o f the angle form edby the vector w i th the posi t ivex-axis.
A
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Magn i tude and direct ion
Consider vector .b,aA 22 baA
a
btan A
Also, .AA sinA,cosAA
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Example.
Solut ion:
Determ ine the magni tude and
direct ion of vectors
and .
44,A 31,B
44,A 22 44 A 32 24
1
4
4 A
tan 1tanArcA
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1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
44,A A
Solut ion (cont inued)
4
1 tanArc
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4
Solut ion (cont inued)
14
4 Atan 1tanArcA
4
3 A
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Solut ion (cont inued)
31
3 Btan 3tanArcB
31,B 22 31 B 4 2
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1 2
1
2
-1-2
-1
-2
B
Solut ion (cont inued)
3tanArcB
31,B
3
B
Also, .3
5 B
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Example.
Solut ion:
Determ ine the components o f
the vector w i th a magn i tude of
6 un i ts in the d irect ion o f .
3
5
AA sinA,cosA
3
5
3
566
sin,cos
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Solut ion (cont inued)
2
36
2
16 ,
AA sinA,cosA 3
5
3
566
sin,cos
333,
5
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1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
Solut ion (cont inued)
3
5
333,6 un i ts
hor izontal componentv
ert
ica
lc
ompone
nt
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Unit vecto r
A un i t vecto r has a
magni tude of 1.
01,i : un i t vector in thedi rect ion of pos i tive
x-axis
10,j : un i t vector in thedi rect ion of pos i tive
y-axis
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Unit vecto r
bjaiA or
Given .b,aA
1001 ,b,aA
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Unit vecto r
Unit vecto r in the d irect ion o f A:
Given .b,aA
A
b,
A
aUA
AA sin,cos
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Example.
Determ ine a un i t vecto r in the
direct ion o f .512 ,Solut ion:
Let .512 ,A
AU
A 22 512 25144
169 13 135
13
12 ,
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I l lustrat ion
2 4 6 8 10-2-2
2
4
6
-4-6-8-10
-4
-6
8
10
512 ,A
AU 135
13
12 ,
-8
-10
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Example.
Determ ine a uni t vecto r in the
direct ion of the vecto r w i th amagn i tude of 10 in the direct ion
of .6
Solut ion:
BU
Let be the g iven vecto r.B
BB sin,cos
66
sin,cos
2
1
2
3,
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VECTORS IN 3D
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The 3D space
The set o f al l ordered tr ipleso f real numbers is called as
the three-dimensional
number space.
3
R z,y,x|z,y,x R
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z axis
xy-planexz-plane
yz-plane
y axis
x axis
z
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-1-2-3-4-5 1 2 3 4 5
1
2
3
4
5
-1
-2
-3
-4
-5
12
34
5
-2-3
-4
-5
x
y
z 432 ,,P
123
,,Q
345 ,,R
Di t d id i t
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Distance and m idpo int
points : 1111 z,y,xP2222 z,y,xP
2
12
2
12
2
12 zzyyxx
Distance: or21PP 21 P,Pd
Di t d id i t
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Distance and m idpo int
Midpoint :
222
212121
21
zz,
yy,
xxM
PP
points : 1111 z,y,xP2222 z,y,xP
E l D t i th d i t
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Example. Determ ine the d istance
between the given points and the
m idpo int of the segment jo in ingthem.
2342 ,,P4231 ,,P
Solut ion:
21PP 2
122
122
12 zzyyxx
222 422334
110362549
S l t i ( t i d)
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Solut ion (cont inued)
2342 ,,P4231 ,,P
222
212121
21
zz,
yy,
xxM PP
2
24
2
32
2
43
21 ,,M PP
12
1
2
1
21 ,,M PP
z
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2342 ,,P
4231 ,,P
1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
12
34
5
-2-3
-4
-5
x
y
z
1
2
1
2
1
21 ,,M PP
V t i 3D
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Vector in 3D
A vecto r in three-d imensional
space is an ordered tr ip le o f
real numbers . a, band bare called components
o f the vecto r.
c,b,a
O th
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On the space
Posi t ion representat ion o f
c,b,a
in i t ial po int : the or ig in
term inal poin t :
000 ,,
c,b,a
M it d d di t i
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Magn i tude and direct ion
Magn i tude o f vector A: A
Direc t ion ang les o f a non-zero
vector A:
smal les t rad ian measu re
measu red from the posi t ive sideo f each axis
z
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x
y
z
MUST!!!
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MUST!!!
In i t ial po int :Term inal po int :
VECTOR COMPONENTS:
ttt z,y,xiii z,y,x
ititit zz,yy,xx
Magn i tude and direct ion
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Magn i tude and direct ion
Consider vector .c,b,aA
222 cbaA
If , and are the d irect ionangles,
A
a
cos
A
b
cos
A
c
cos
1222 coscoscoswhere
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Unit vectors
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Unit vectors
001 ,,i : un i t vecto r in thedi rect ion of pos i tive
x-axis010
,,j : un i t vector in thedi rect ion of pos i tivey-axis
100 ,,k : un i t vecto r in thedi rect ion of pos i tive
z-axis Unit vecto r
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Unit vecto r
ckbjaiA
Given .
c,b,aA
Ac,
Ab,
AaUA
cos,cos,cos
E l
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Consider vector w i th in i t ial
poin t at and term inal
po int at . 542 ,, 314 ,,
Example.
A
Determ ine the fol low ing :i. the components of the vector ;
i i . i ts magn i tude and direct ional
cos ines; andii i . the un i t vecto r in the same
direct ion
Solut ion:
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Solut ion:
542 ,,: tP 314 ,,:Pi
components of the vector : 256 ,,magni tude: 65
direct ion cos ines:
65
6cos
65
5cos
65
2cos
un i t vecto r in the
same direct ion : 65
2
65
5
65
6,,
z
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1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
12
34
5
-2-3
-4
-5
x
y
256 ,,
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END