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Section 9.1Polar Coordinates
xOrigin PolePolar axis
Polar axis
r
P r ,
O Pole
The Polar Plane Coordinates (r, θ)
0
15
30
45
607590105
120
135
150
165
180
195
210
225
240255 270 285
300
315
330
345
Polar axisO Pole
6
4
6,4
P
s.coordinatepolar using 6
4,point Plot the
r
P r r , , 0
Plotting r r, , 0
s.coordinatepolar using 6
7,4point Plot the
4
6
7,4
P
O
76
6
7
Find other polar coordinates of the point
2, 3 for which
(a) (b)
c)
r
r r
r
,
, ,
( ,
0 2 4 0 0 2
0 2 0
( , ,a) 7 3P 2 3 2 2
( , ,b) 4 3P 2 3 2
( , ,c) 5 3P 2 3 2 2 8
Section 9.2Polar Equations and Graphs
9
Identify and graph the equation: r = 2r 2 r2 4 x y2 2 4
Circle with center at the pole and radius 2.
0
15
30
45
607590105
120
135
150
165
180
195
210
225
240255 270 285
300
315
330
345
43210
Identify and graph the equation: =3
3tantan
31
yx 3
1y x 3
0
15
30
45
607590105
120
135
150
165
180
195
210
225
240255 270 285
300
315
330
345
43210
3
Identify and graph the equation: r sin 2
sin sin ryr
y y 2
0
15
30
4560
7590105120
135
150
165
180
195
210
225240
255 270 285300
315
330
34543210
Let a be a nonzero real number, the graph of the equation
r asin
is a horizontal line y = a
Let a be a nonzero real number, the graph of the equation
r acos
is a vertical line x = a
Identify and graph the equation: r 4 cos
r r2 4 cos
x y x2 2 4
x x y2 24 0
x x y2 24 4 4
x y 2 42 2
15
0
15
30
4560
7590105120
135
150
165
180
195
210
225240
255 270 285300
315
330
34543210
16
Let a be a positive or negative real number. Then,
r a2 sin Circle: radius a ; center at (0, a)
r a2 cos Circle: radius a ; center at (a, 0).
17
Symmetry with Respect to the Polar Axis (x-axis):
0
15
30
45
607590105
120
135
150
165
180
195
210
225
240255 270 285
300
315
330
345
r ,
r ,
18
Symmetry with Respect to the Line (y-axis) 2
0
15
30
45
607590105
120
135
150
165
180
195
210
225
240255 270 285
300
315
330
345
r , r ,
Symmetry with Respect to the Pole (Origin):
0
15
30
45
607590105
120
135
150
165
180
195
210
225
240255 270 285
300
315
330
345
r ,
r ,
Tests for Symmetry
Symmetry with Respect to the Polar Axis (x-axis): Replace θ by - θ
Symmetry with Respect to the Line (y-axis): Replace θ by Π - θ
2
Symmetry with Respect to the Pole (Origin): Replace r by -r
If an equivalent equation results then the graph is symmetric with respect to the given pole or line.
21
Specific Types of Polar Graphs
Cardioids (heart shaped)
sin1cos1
sin1)cos1(
arar
arar
where a > 0. The graph passes through the pole.
Limacons without an inner loop (French word for snail)
sincos
sincos
barbar
barbar
where a > 0, b > 0, and a > b. The graph does not pass through the pole.
22
Limacons with an inner loop (French word for snail)
sincos
sincos
barbar
barbar
where a > 0, b > 0, and a < b. The graph passes through the pole twice.
Rose Curves
narnar sincos
If n is even and not zero, the graph has 2n petals. If n is odd and not one or negative one, the graph has n petals.
Lemniscates (Greek word for propeller)
2sin2cos 2222 arar
where a is non-zero. The graph will be propeller shaped.
24
Section 9.3The Complex Plane
25
Real Axis
Imaginary Axis
O
z x yi
The Complex Plane
Real Axis
Imaginary Axis
O
z x yi
z
x
y
z x y 2 2
z is the magnitude of z = x + yi27
z x yi r r i cos sin Cartesian
Form
Polar Form
z r i cos sin
z r
AND
28
Plot the point corresponding to in
the complex plane, and write an expression
for in polar form.
z i
z
3 4
4
-3 Real Axis
Imaginary Axisz i 3 4
29
z i 3 4 Quadrant II
x y 3 4 and
r ( )3 4 9 16 52 2
sin yr
45
0 < 2
9.1261.53180
9.126sin9.126cos5sincos iirz
Write an expression for
in rectangular form.
z i 3 330 330cos sin
z i 3 330 330cos sin
i
2
1
2
33
DeMoivre’s Theorem
If is a complex number,
then
z r i cos sin
z r n i nn n cos sin
integer. positive a is 1 where n
32
Write in the
standard form
3 30 304
cos sin
.
i
a bi
3 30 304
cos sin i
304sin304cos34 i
81 120 120cos sin i
ii2
381
2
81
2
3
2
181
Write in the standard form 34
i a bi.
r 3 1 4 22 2
ii
2
1
2
323
6
5sin
6
5cos2
i
4
4
6
5sin
6
5cos23
ii
6
54sin
6
54cos24
i
34
i
6
54sin
6
54cos24
i
3
10sin
3
10cos16
i
i
2
3
2
116
8 8 3i
35
Section 9.4Vectors
36
A vector is a quantity that has both magnitude and direction.
Vectors in the plane can be represented by arrows.
The length of the arrow represents the magnitude of the vector.
The arrowhead indicates the direction of the vector.
P
Q
Initial Point
Terminal Point
PQv
The vector v whose magnitude is 0 is called the zero vector, 0.
v w if they have the same magnitude and direction.
Two vectors v and w are equal, written
vw
v w
v
wv w
Initial point of v
Terminal point of w
Vector Addition
Vector addition is commutative.
v w w v Vector addition is associative.
u v w u v w
v 0 0 v v
v v 0
Properties of Scalar Products
0 1 1v 0 v v v v
v v v v w v w
v v
42
Use the vectors illustrated below to graph each expression.
v
w
u
43
v w
v w
wv - and 2
v
2v
w
w
2v w
2v
w
46
If is a vector, we use the symbol to
represent the of
v v
magnitude v.
vv
vv
0vv
v
v
(d)
(c)
ifonly and if 0 b)(
0 (a)
thenscalar, a is if and vector a is If
47
A vector for which is called a
.
u u
unit vector
1
Let i be a unit vector along the pos. x-axis;
Let j be a unit vector along the pos. y-axis.
If v has initial point at the origin O and terminal point at P = (a, b), then
v i j a b
48
ai
bj
a
P = (a, b)
v = ai
+ bjb
The scalars a and b are called components of the vector v = ai + bj.
v i j x x y y2 1 2 1
Position Vector
The position vector re-positions the vector so that the initial point is the origin.
50
.4,3 and 1,2 if
vector theofector position v theFind
2121
PPPPv
v i j x x y y2 1 2 1
v i j 3 2 4 1( )
v i j 5 3
P1 2 1 ,
P2 3 4 ,
5 3,
O
v = 5i + 3j
52
Section 9.6Vectors in Space
53
In space, each point is associated with an ordered triple of real numbers. Through a fixed point, the origin, O, draw three mutually perpendicular lines, the x-axis, y-axis, and z-axis.
z
y
x
2-2
2
-22
-2
O
Distance Formula in Space
If and are
two points in space, the distance from
to is
P x y z P x y z
d
P P
1 1 1 1 2 2 2 2
1 2
, , , ,
d x x y y z z 2 12
2 12
2 12
Find the distance from
to
P
P1
2
1 5 3
6 4 1
, ,
, ,
d 6 1 4 5 1 32 2 2
49 81 4
134
56
If v is a vector with initial point at the origin O and terminal point at P = (a, b, c), then we can represent v in terms of the vectors i, j, and k as
v = ai + bj + ck
P = (a, b, c)
v = ai + bj + ck
Position Vector 59
Suppose that is a vector with initial point
, not necessarily the origin,
and terminal point If
then is equal to the position vector
v
v v
P x y z
P x y z
P P
1 1 1 1
2 2 2 2
1 2
, ,
, , .
,
v i j k x x y y z z2 1 2 1 2 1
P x y z1 1 1 1 , , P x y z2 2 2 2 , ,
v
P P1 2
x x y y z z2 1 2 1 2 1 i j k
Properties of Dot Product
If u, v, and w are vectors, thenCommutative Property
u v v u
Distributive Property
u v w u v u w
v v v
0 v
2
059
Section 9.7The Cross Product
If v = a1i + b1j + c1k and w = a2i + b2j + c2k are two vectors in space, the cross product v x w is defined as the vector
v x w = (b1c2 - b2c1)i - (a1c2 - a2c1)j + (a1b2 - a2b1)k
Example: If v = 3i + 2j + 4k and w = 2i + j + 2k, find the cross product v x w.
kjiwv )2213()4223()4122(
kji )43()86()44(
kj 2
If a, b, c, and d are four real numbers, thesymbol
Da b
c d
is called a 2 by 2 determinant. Its value isthe number ad - bc; that is,
D
a b
c dad bc
A 3 by 3 determinant is symbolized by
222
111
cba
cba
CBA
22
11
22
11
22
11
ca
baC
ca
caB
cb
cbA
2
3 1
2 31
2 1
4 31
2 3
4 2( )
324
132
112
)124(1)46(1292
)16(1)2(1112 36
Evaluate the following determinant.
Determinates can be used to find cross products. Find v x w is v = 3i + 2j + 4k and w = 2i + j + 2k.
212
423
kji
wv
kji12
23
22
43
21
42
kji )43()86()44(
kj 2
If u, v, and w are vectors in space and if a is a scalar, then
u x u = 0
u x v = -(v x u)
a(u x v) = (au) x v = u x (av)
u x (v x w) = (u x v) + (u x w)
Algebraic Properties of the Cross Product
If u, v, and w are vectors in space and if a is a scalar, then
u x v is orthogonal to both u and v
u x v=0 if and only if u and v are orthogonal
Geometric Properties of the Cross Product
Find a vector that is orthogonal to u = 2i - 3j + k and v = -3i - j + k.
113
132
kji
vu
13
32
13
12
11
13
ji
kji 923213
kji 752
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