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