Ch.10 PolarCoordinates;Vectors
10.1 PolarCoordinates
1 PlotPointsUsingPolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
MatchthepointinpolarcoordinateswitheitherA,B,C,orDonthegraph.
1) –3,π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
A) A B) B C) C D) D
2) 3,–5π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
A) A B) B C) C D) D
Page1
3) –3,–π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
A) A B) B C) C D) D
4) 3,π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
5) 3,–π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
Page2
6) –3,–π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
7) –3,4π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
8) –3,5π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
Page3
Plotthepointgiveninpolarcoordinates.
9) 2,–π
4
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page4
10) 2,7π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
11) –2,–5π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page6
12) 2,–3π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
13) –4,5π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
14) (5
,
30°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
15) (4
,
–225°)
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page10
16) (–4
,
405°)
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page11
17) (–2,45°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page12
18) (2,45°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
19) 3,7π
6
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
20) (2,0°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
21) (2,360°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page16
22) –3,–π
4
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
23) 5,5π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
24) Plotthepoint4,π
6
andfindotherpolarcoordinates(r,θ)ofthepointforwhich:
(a) r>0,–2π≤θ<0
(b) r<0,0≤θ<2π
(c) r>02π≤θ<4π
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
25) Plotthepoint4,5π
6
andfindotherpolarcoordinates(r,θ)ofthepointforwhich:
(a) r>0,–2π≤θ<0
(b) r<0,0≤θ<2π
(c) r>02π≤θ<4π
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page19
2 ConvertfromPolarCoordinatestoRectangularCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Thepolarcoordinatesofapointaregiven.Findtherectangularcoordinatesofthepoint.
1) 3,2π
3
A) –3
2,33
2B) 3
2,33
2C) –3
2,–33
2D) 3
2,–33
2
2) –7,2π
3
A) 7
2,–73
2B) –7
2,–73
2C) 7
2,73
2D) –7
2,73
2
3) –5,3π
4
A) 52
2,–52
2B) –52
2,52
2C) –52
2,–52
2D) 52
2,52
2
4) 5,3π
4
A) –52
2,52
2B) 52
2,–52
2C) 52
2,52
2D) –52
2,–52
2
5) 5,–4π
3
A) –5
2,53
2B) 5
2,–53
2C) 53
2,5
2D) –53
2,–5
2
6) (–7
,
120°)
A) 7
2,–73
2B) –7
2,–73
2C) 7
2,73
2D) –7
2,73
2
7) (–3
,
–135°)
A) 32
2,32
2B) 32
2,–32
2C) –32
2,–32
2D) –32
2,32
2
8) (–4
,
–180°)
A) (4
,
0) B) (0
,
4) C) (–4
,
0) D) (0
,
–4)
9) (400,130°)Roundtherectangularcoordinatestotwodecimalplaces.
A) (–257.12,306.42) B) (–257.12,–306.42) C) (306.42,–257.12) D) (306.42,257.12)
10) (4,70°)Roundtherectangularcoordinatestotwodecimalplaces.
A) (1.37,3.76) B) (3.76,1.37) C) (1.59,4.01) D) (4.01,1.59)
11) (8.9
,
6.5)Roundtherectangularcoordinatestotwodecimalplaces.
A) (8.69
,
1.91) B) (8.84
,
1.01) C) (8.84
,
–1.01) D) (8.69
,
–1.91)
3 ConvertfromRectangularCoordinatestoPolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Therectangularcoordinatesofapointaregiven.Findpolarcoordinatesforthepoint.
1) (–6
,
0)
A) (6
,
π)B)6,3π
2C) (–6
,
π)D)6,π
2
2) (0,5)
A) 5, π
2B) 5,–π
2C) (5
,
0) D) (5
,
π)
3) (–5
,
5)
A) 5 2,3π
4B) 5 2,–3π
4C) –52,π
4D) –52,–3π
4
4) ( 3,–1)
A) 2,–π
6B) 2, π
6C) 2,–5π
6D) 2, 5π
6
5) (–3
,
2.5)Roundthepolarcoordinatestotwodecimalplaces,withθ inradians.
A) (3.91
,
2.45) B) (3.91
,
0.88) C) (3.91
,
–0.88) D) (–3.91
,
0.88)
6) (100,–30)Roundthepolarcoordinatestotwodecimalplaces,withθ indegrees.
A) (104.40,–16.70°) B) (104.40,16.70°) C) (104.40,106.70°) D) (104.40,–106.70°)
7) (0.6,–1.1)Roundthepolarcoordinatestotwodecimalplaces,withθ indegrees.
A) (1.25,–61.39°) B) (1.25,61.39°) C) (1.25,–57.93°) D) (1.25,57.93°)
Solvetheproblem.
8) Awomanwalks100yardseastwardalongastraightshorelineandthenswims30yardssouthwardinto
theoceanonalinethatisperpendiculartotheshoreline.Usingherstartingpointasthepoleandtheeast
directionasthepolaraxis,givehercurrentpositioninpolarcoordinates.Roundthecoordinatestothe
nearesthundredth.Expressθindegrees.
A) (104.40,–16.70°) B) (104.40,–88.28°) C) (11.40,–16.70°) D) (11.40,–88.28°)
9) Afiretruckisenroutetoanaddressthatis6blockseastand11blockssouthofthefirestation.Usingthe
firestationasthepoleandtheeastdirectionasthepolaraxis,expressthefiretruckʹsdestinationinpolar
coordinates.Roundthecoordinatestothenearesthundredth.Expressθindegrees.
A) (12.53,–61.39°) B) (12.53,–28.61°) C) (4.12,–61.39°) D) (4.12,–28.61°)
4 TransformEquationsbetweenPolarandRectangularForms
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Thelettersxandyrepresentrectangularcoordinates.Writetheequationusingpolarcoordinates(r,θ).
1) x2+4y2=4
A) r2(cos2θ+4sin2θ)=4B)r
2(4cos2θ+sin2θ)=4
C) cos2θ+4sin2θ=4r D) 4cos2θ+sin2θ=4r
2) x2+y2–4x=0
A) r=4cosθ B) r=4sinθ C) rcos2θ=4sinθ D) rsin2θ=4cosθ
Page21
3) x2=4y
A) rcos2θ=4sinθ B) rsin2θ=4cosθ C) 4cos2θ=rsinθ D) 4sin2θ=rcosθ
4) y2=16x
A) rsin2θ=16cosθ B) r2sin2θ=16cosθ
C) sin2θ=16rcosθ D) sin2θ=16r2cosθ
5) xy=1
A) r2sin2θ=2B)2rsinθ cosθ =1C)rsin2θ=2D)2r
2sinθcosθ=1
6) 2xy=1
A) r2sin2θ=1B)2rcosθ sinθ =1C)r
2sin2θ=2D)r
2cosθsinθ=2
7) 2x+3y=6
A) r(2cosθ+3sinθ)=6B) r(2sinθ +3cosθ)=6
C) 2cosθ+3sinθ=6r D) 2sinθ +3cosθ =6r
8) x=–3
A) rcosθ=–3B)rcosθ =3C)rsinθ =3D)rsinθ = –3
9) y=5
A) rsinθ=5B)rcosθ =5C)r=5 D) sinθ cosθ =5
10) y=x
A) sinθ=cosθ B) r=sinθ C) r=cosθ D) sinθ = – cosθ
Thelettersrandθrepresentpolarcoordinates.Writetheequationusingrectangularcoordinates(x,y).
11) r=cosθ
A) x2+y2=xB)x
2+y2=yC)(x+y)2=xD)(x+y)2=y
12) r=1+2sinθ
A) x2+y2=x2+y2+2y B) x2+y2=x2+y2+2x
C) x2+y2=x2+y2+2y D) x2+y2=x2+y2+2x
13) r=10sinθ
A) x2+y2=10y B) x2+y2=10x C) x2+y2=10y D) x2+y2=10x
14) r=2(sinθ–cosθ)
A) x2+y2=2y–2x B) 2x2+2y2=y–xC)x
2+y2=2x–2y D) 2x2+2y2=x–y
15) r=5
A) x2+y2=25 B) x2–y2=25 C) x+y=25 D) x+y=5
16) r=5
1+cosθ
A) y2=25–10x B) y2=10x–25 C) x2=25–10y D) x2=10y–25
17) rsinθ=10
A) y=10 B) x=10 C) y=10x D) x=10y
Page22
18) r(1–2cosθ)=1
A) x2+y2=1+2x B) x2+y2=1+2x C) x2+y2=2+xD)x
2+y2=2+x
Page23
10.2 PolarEquationsandGraphs
1 IdentifyandGraphPolarEquationsbyConvertingtoRectangularEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Transformthepolarequationtoanequationinrectangularcoordinates.Thenidentifyandgraphtheequation.
1) r=5
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2+y2=25;circle,radius5,
centeratpole
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2+y–5
2
2=25
4;circle,
radius5
2,centerat0,5
2
in
rectangularcoordinates
C)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x–5
2
2+y2=25
4;circle,
radius5
2,centerat5
2,0in
rectangularcoordinates
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x=5;verticalline5unitstotheright
ofthepole
2) r=4sinθ
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2+(y–2)2=4;circle,radius2,
centerat(0,2)inrectangularcoordinates
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x–2)2+y2=4;circle,radius2,
centerat(2,0)inrectangularcoordinates
Page25
C)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2+(y+2)2=4;circle,radius2,
centerat(0,–2)inrectangularcoordinates
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x+2)2+y2=4;circle,radius2,
centerat(–2,0)inrectangularcoordinates
3) r=2cosθ
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x–1)2+y2=1;circle,radius1,
centerat(1,0)inrectangularcoordinates
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2+(y–1)2=1;circle,radius1,
centerat(0,1)inrectangularcoordinates
Page26
C)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2+(y+1)2=1;circle,radius1,
centerat(0,–1)inrectangularcoordinates
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x+1)2+y2=1;circle,radius1,
centerat(–1,0)inrectangularcoordinates
4) rsinθ=4
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
y=4;horizontalline4units
abovethepole
B)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x=4;verticalline4units
totherightofthepole
Page27
C)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x=–4;verticalline4units
totheleftofthepole
D)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
y=–4;horizontalline4units
belowthepole
5) θ=π
3
r
-5 5
5
-5
r
-5 5
5
-5
A)
r
-5 5
5
-5
r
-5 5
5
-5
y=3x;linethroughthepolemaking
anangleofπ
3
withthepolaraxis
B)
r
-5 5
5
-5
r
-5 5
5
-5
y=–π
3;horizontallineπ
3
units
belowthepole
C)
r
-5 5
5
-5
r
-5 5
5
-5
y=–3
3x;linethroughthepolemaking
anangleofπ
3
withthepolaraxis
D)
r
-5 5
5
-5
r
-5 5
5
-5
x–π
3
2+y2=π2
9;circle,radiusπ
3,
centeratπ
3,0inrectangularcoordinates
6) rsecθ=–6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x+3)2+y2=9;circle,radius3
center(–3,0)inrectangularcoordinates
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x=–6;verticalline6units
totheleftofthepole
Page29
C)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
y=–6;horizontalline6units
belowthepole
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2+(y+3)2=9;circle,radius3,
centerat(0,–3)inrectangularcoordinates
Matchthegraphtooneofthepolarequations.
7)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=2B)r=4 cosθ C) r=4 sinθ D) rsinθ =2
8)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A) θ=π
6B) θ=–π
6C) r=π
6D) r=–π
6
9)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=–2sinθ B) r=–2 cosθ C) r= –1D)rsinθ = –1
10)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=4cosθ B) r=4 sinθ C) r=2D)rsinθ =2
11)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) rsinθ=1B)r=2 cosθ C) r=1D)r=2 sinθ
12)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=2+sinθ B) r=4 cosθ C) r=4 sinθ D) r=2+cosθ
13)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r=4+cosθ B) r=8 cosθ C) r=8 sinθ D) r=4+sinθ
2 TestPolarEquationsforSymmetry
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Testtheequationforsymmetrywithrespecttothegivenaxis,line,orpole.
1) r=–4cosθ;thepolaraxis
A) Symmetricwithrespecttothepolaraxis
B) Mayormaynotbesymmetricwithrespecttothepolaraxis
2) r=4cosθ;thelineθ=π
2
A) Mayormaynotbesymmetricwithrespecttothelineθ=π
2
B) Symmetricwithrespecttothelineθ=π
2
3) r=–4sinθ;thepole
A) Mayormaynotbesymmetricwithrespecttothepole
B) Symmetricwithrespecttothepole
4) r=4+4sinθ;polaraxis
A) Mayormaynotbesymmetricwithrespecttothepolaraxis
B) Symmetricwithrespecttothepolaraxis
5) r=3–3cosθ;thelineθ=π
2
A) Mayormaynotbesymmetricwithrespecttothelineθ=π
2
B) Symmetricwithrespecttothelineθ=π
2
6) r=4+2sinθ;thelineθ=π
2
A) Symmetricwithrespecttothelineθ=π
2
B) Mayormaynotbesymmetricwithrespecttothelineθ=π
2
7) r=4+2cosθ;thepole
A) Mayormaynotbesymmetricwithrespecttothepole
B) Symmetricwithrespecttothepole
8) r=3–6sinθ;thepolaraxis
A) Mayormaynotbesymmetricwithrespecttothepolaraxis
B) Symmetricwithrespecttothepolaraxis
9) r2=sin(2θ);thepole
A) Symmetricwithrespecttothepole
B) Mayormaynotbesymmetricwithrespecttothepole
10) r=2sin(3θ);thelineθ=π
2
A) Symmetricwithrespecttothelineθ=π
2
B) Mayormaynotbesymmetricwithrespecttothelineθ=π
2
3 GraphPolarEquationsbyPlottingPoints
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Identifyandgraphthepolarequation.
1) r=1–sinθ
r
-5 5
5
-5
r
-5 5
5
-5
A)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
B)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
C)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
D)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
2) r=4+3sinθ
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
A)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithoutinnerloop
B)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithoutinnerloop
C)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithinnerloop
D)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithinnerloop
3) r=5+6cosθ
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
A)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithinnerloop
B)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithinnerloop
C)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithoutinnerloop
D)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limaconwithoutinnerloop
4) r=4sin(2θ)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
rosewithfourpetals
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
rosewithtwopetals
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
circle
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
lemniscate
Page37
5) r2=4sin(2θ)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
A)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
lemniscate
B)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
lemniscate
C)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
rosewithfourpetals
D)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
rosewithfourpetals
Page38
6) r=3θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmicspiral
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmicspiral
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmicspiral
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmicspiral
Thepolarequationofthegraphiseitherr=a+bcosθorr=a+bsinθ,a>0,b>0.Matchthegraphtooneofthe
equations.
7)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r=3+4cosθ B) r=4+3 cosθ C) r=3+4 sinθ D) r=4+3 sinθ
8)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r=4+3cosθ B) r=3+4 cosθ C) r=3+4 sinθ D) r=4+3 sinθ
9)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r=4+3sinθ B) r=3+4 cosθ C) r=3+4 sinθ D) r=4+3 cosθ
10)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r=2+3sinθ B) r=2+3 cosθ C) r=3+2 sinθ D) r=3+2 cosθ
Graphthepolarequation.
11) r=1+cosθ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
12) r=2
1–cosθ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
13) r=3
1–sinθ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
14) r=2
1–2sinθ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
15) r=3
3–2cosθ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page46
16) r=4
4–2sinθ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
17) r=θ
,
θ≥0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page48
18) r=4
θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page49
19) r=cscθ–2
,
0
<
θ
<
π
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
20) r=sinθtanθ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page51
21) r=tanθ,–π
2
<θ<π
2
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
10.3 TheComplexPlane;DeMoivreʹsTheorem
1 PlotPointsintheComplexPlane
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Plotthecomplexnumberinthecomplexplane.
1) –1+6i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
2) 2i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
3) –4+i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page55
4) –6–i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
5) 3+22i
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
A)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
B)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
C)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
D)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
Writethecomplexnumberinrectangularform.
6) 8 cosπ
6
+isinπ
6
A) 4 3+4i B) 4+43iC)
3
4
+1
4iD)
1
4
+3
4i
7) 3 cosπ
3
+isinπ
3
A) 3
2
+33
2iB)
3
2
+33
2iC)3+iD)
3
6
+3
6i
8) 4 cos11π
6
+isin11π
6
A) 2 3–2i B) –23–2i C) 2+23iD)2–23i
9) 4(cos300°+isin300°)
A) 2–23
iB)
–2+23iC)23–2i D) –23–2i
10) 6(cos330°+isin330°)
A) 3 3–3i B) 3 3+3i C) –33+3i D) –33–3i
11) 9(cos180°+isin180°)
A) –9B)9 C)
–9i D) 9i
2 ConvertaComplexNumberbetweenRectangularFormandPolarForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writethecomplexnumberinpolarform.Expresstheargumentindegrees,roundedtothenearesttenth,if
necessary.
1) 3–i
A) 2(cos330°+isin330°) B) 2(cos300° +isin300°)
C) 4(cos330°+isin330°) D) 4(cos300° +isin300°)
2) 2+2i
A) 2 2(cos45°+isin45°) B) 4(cos45° +isin45°)
C) 2 2(cos30°+isin30°) D) 4(cos30° +isin30°)
3) –2
A) 2(cos180°+isin180°) B) 2(cos0° +isin0°)
C) 2(cos270°+isin270°) D) 2(cos90° +isin90°)
4) 5i
A) 5(cos90°+isin90°) B) 5(cos270° +isin270°)
C) 5(cos0°+isin0°) D) 5(cos180° +isin180°)
5) –12+16i
A) 20(cos126.9°+isin126.9°) B) 20(cos53.1° +isin53.1°)
C) 20(cos233.1°+isin233.1°) D) 20(cos306.9° +isin306.9°)
3 FindProductsandQuotientsofComplexNumbersinPolarForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findzworz
wasspecified.Leaveyouranswerinpolarform.
1) z=10(cos30°+isin30°)
w=5(cos10°+isin10°)
Findzw.
A) 50(cos40°+isin40°) B) 50(cos300° +isin300°)
C) 15(cos300°+isin300°) D) 15(cos40° +isin40°)
Page58
2) z=10(cos45°+isin45°)
w=5(cos15°+isin15°)
Findzw.
A) 50(cos60°+isin60°) B) 5(cos30° +isin30°)
C) 5(cos60°+isin60°) D) 50(cos30° +isin30°)
3) z=5(cos35°+isin35°)
w=2(cos40°+isin40°)
Findzw.
A) 10(cos75°+isin75°) B) 7(cos75° +isin75°)
C) 10(cos50.9°+isin50.9°) D) 7(cos50.9° +isin50.9°)
4) z=8 cosπ
6
+isinπ
6
w=3 cosπ
2
+isinπ
2
Findzw.
A) 24 cos2π
3
+isin2π
3B) 12 cos2π
3
+isin2π
3
C) 24 cosπ
3
+isinπ
3D) 12 cosπ
3
+isinπ
3
5) z=6 cos3π
2
+isin3π
2
w=12 cos5π
6
+isin5π
6
Findzw.
A) 72 cosπ
3
+isinπ
3B) 36 cosπ
3
+isinπ
3C) 72 cosπ
6
+isinπ
6D) 36 cosπ
6
+isinπ
6
6) z=2+2i
w=3–i
Findzw.
A) 4 2 cosπ
12
+isinπ
12 B) 4 2 cos23π
12
+isin23π
12
C) 4 cosπ
12
+isinπ
12 D) 4 cos23π
12
+isin23π
12
7) z=10(cos30°+isin30°)
w=5(cos10°+isin10°)
Findz
w.
A) 2(cos20°+isin20°) B) 5(cos3° +isin3°)
C) 5(cos20°+isin20°) D) 2(cos3° +isin3°)
Page59
8) z=10(cos45°+isin45°)
w=5(cos15°+isin15°)
Findz
w.
A) 2(cos30°+isin30°) B) 2(cos45° +isin45°)
C) 1
2(cos30°+isin30°) D) 1
2(cos45°+isin45°)
9) z=5(cos200°+isin200°)
w=4(cos50°+isin50°)
Findz
w.
A) 5
4(cos150°+isin150°) B) 5
4(cos40°+isin40°)
C) 4
5(cos150°+isin150°) D) 4
5(cos40°+isin40°)
10) z=8 cosπ
2
+isinπ
2
w=3 cosπ
6
+isinπ
6
Findz
w.
A) 8
3cosπ
3
+isinπ
3B) 8
3cosπ
12
+isinπ
12
C) 5 cosπ
3
+isinπ
3D) 5 cosπ
12
+isinπ
12
11) z=3 cos7π
4
+isin7π
4
w=6 cos9π
4
+isin9π
4
Findz
w.
A) 2
2cos3π
2
+isin3π
2B) 2
2cosπ
2
+isinπ
2
C) 3 2 cos3π
2
+isin3π
2D) 3 2 cosπ
2
+isinπ
2
Page60
12) z=6 cos3π
2
+isin3π
2
w=12 cos5π
6
+isin5π
6
Findz
w.
A) 1
2cos2π
3
+isin2π
3B) 1
2cosπ
3
+isinπ
3
C) 1
6cos2π
3
+isin2π
3D) 1
6cosπ
3
+isinπ
3
13) z=1–i
w=1–3i
Findz
w.
A) 2
2(cos15°+isin15°) B) 1
2(cos75°+isin75°)
C) 1
2(cos15°+isin15°) D) 2
2(cos75°+isin75°)
14) z=1+i
w=3–i
Findzw.
A) 2 2(cos15°+isin15°) B) 2
2(cos15°+isin15°)
C) 2 2(cos75°+isin75°) D) 2 2(cos345°+isin345°)
15) z=1+i
w=1–3i
Findz
w.
A) 2
2(cos105°+isin105°) B) 1
2(cos15°+isin15°)
C) 1
2(cos105°+isin105°) D) 2
2(cos15°+isin15°)
4 UseDeMoivreʹsTheorem
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writetheexpressioninthestandardforma+bi.
1) 2(cos15°+isin15°) 3
A) 4 2+42iB)32+32iC)4+4i D) 3+3i
2) 2(cos75°+isin75°) 3
A) –42
–42iB)42–42iC)
–4–42iD)42+42i
Page61
3) 2(cos105°+isin105°) 3
A) 4 2–42iB)
–42–42iC)4–42iD)
–42+42i
4) 2 cos3π
4
+isin3π
4
4
A) –4B)4 C)
–4i D) 4i
5) 3 cos5π
6
+isin5π
6
4
A) –9
2
–93
2iB)
–93
2
–9
2iC)
–9
2
+93
2iD)
–93
2
+9
2i
6) (1+i)20
A) –1024 B) 1024i C) –1024i D) 1024
7) (1–i)10
A) –32i B) 32 C) 32–32i D) –32+32i
8) (–3+i)6
A) –64 B) 64i C) –64 3+64i D) 64–64 3i
9) ( 3+i)5
A) –16 3+16i B) 9 3+5i C) 16–16 3iD)163–16i
10) –1
2
–3
2i10
A) –1
2
–3
2iB)
–1
2
+3
2iC)
1
2
+3
2iD)
1
2
–3
2i
5 FindComplexRoots
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findallthecomplexroots.Leaveyouranswersinpolarformwiththeargumentindegrees.
1) Thecomplexcuberootsof–8i
A) 2(cos90°+isin90°),2(cos210°+isin210°),2(cos330°+isin330°)
B) 8(cos90°+isin90°),8(cos210°+isin210°),8(cos330° +isin330°)
C) 512(cos90°+isin90°),512(cos210° +isin210°),512(cos330° +isin330°)
D) 2(cos180°+isin180°),2(cos300° +isin300°),2(cos60° +isin60°)
2) Thecomplexfourthrootsof–16
A) 2(cos45°+isin45°),2(cos135°+isin135°),2(cos225° +isin225°),16(cos315°+isin315°)
B) 42(cos45°+isin45°),42(cos135°+isin135°),42(cos225°+isin225°),42(cos315°+isin315°)
C) 16(cos45°+isin45°),16(cos135° +isin135°),16(cos225° +isin225°),16(cos315°+isin315°)
D) 2(cos90°+isin90°),2(cos180°+isin180°),2(cos270° +isin270°),2(cos360°+isin360°)
Page62
3) Thecomplexfifthrootsof3+i
A) 52(cos6°+isin6°),52(cos78°+isin78°),52(cos150°+isin150°),52(cos222°+isin222°),
52(cos294°+isin294°)
B) 32(cos6°+isin6°),32(cos78°+isin78°),32(cos150° +isin150°),32(cos222°+isin222°),
32(cos294°+isin294°)
C) 52(cos30°+isin30°),52(cos102°+isin102°),52(cos174°+isin174°),52(cos246°+isin246°),
52(cos318°+isin318°)
D) 32(cos30°+isin30°),32(cos102° +isin102°),32(cos174° +isin174°),32(cos246°+isin246°),
32(cos318°+isin318°)
4) Thecomplexfifthrootsof–2i
A) 52(cos54°+isin54°),52(cos126°+isin126°),52(cos198°+isin198°),52(cos270°+isin270°),
52(cos342°+isin342°)
B) 52(cos45°+isin45°),52(cos117°+isin117°),52(cos189°+isin189°),52(cos261°+isin261°),
52(cos333°+isin333°)
C) 42(cos54°+isin54°),42(cos126°+isin126°),42(cos198°+isin198°),42(cos270°+isin270°),
42(cos342°+isin342°)
D) 32(cos54°+isin54°),32(cos126° +isin126°),32(cos198° +isin198°),32(cos270°+isin270°),
32(cos342°+isin342°)
10.4 Vectors
1 GraphVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usethevectorsinthefigurebelowtographthefollowingvector.
1) u+z
A) B)
C) D)
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2) 3w
A) B)
C) D)
3) v–w
A) B)
C) D)
4) z–v
A) B)
C) D)
5) 2u–z–w
A) B)
C) D)
Usethefigurebelow.Determinewhetherthegivenstatementistrueorfalse.
6) A+H=F
A) True B) False
7) G+H=F
A) True B) False
8) H+I+J=B
A) True B) False
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9) A+B+C+D+E=0
A) True B) False
10) C+D+G+I+J=0
A) True B) False
2 FindaPositionVector
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
ThevectorvhasinitialpositionPandterminalpointQ.Writevintheformai+bj;thatis,finditspositionvector.
1) P=(0,0);Q=(2
,
–5)
A) v=2i–5jB) v=–5i–5jC) v= –2i+5jD) v=5i–2j
2) P=(2
,
6);Q=(–1
,
–2)
A) v=–3i–8jB) v=–8i–3jC) v=3i+8jD) v=8i+3j
3) P=(4
,
5);Q=(–5
,
1)
A) v=–9i–4jB) v=–4i–9jC) v= –10i–3jD) v= –3i–10j
3 AddandSubtractVectorsAlgebraically
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Ifu=11i–5jandv=–3i+8j
,
findu+v.
A) 8i+3jB) 7i+3jC) 14i+7jD) –14i+3j
2) Ifu=9i–2jandv=–3i+7j
,
findu–v.
A) 12i–9jB) 6i+5jC) 11i+5jD) 10i+5j
4 FindaScalarMultipleandtheMagnitudeofaVector
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Ifw=6i+2j
,
find4w.
A) 24i+8jB) 24i+2jC) 10i+6jD) 10i+2j
2) Ifv=3i–5jandw=–7i+4j,find3v–4w.
A) 37i–31jB) –4i–jC) –19i+jD) 17i–10j
3) Ifv=9i+12j
,
findv.
A) 15 B) 225 C) 21 D) 15
4) Ifv=–4i+4j
,
findv.
A) 4 2 B) 2 2 C) 8 D) 32
5) Ifv=–i–j
,
findv.
A) 2 B) 2 C) 0 D) 1
6) Ifv=7i+8j
,
whatis–10v?
A) 10 113 B) 10i 15 C) 10 15 D) –10 113
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7) Ifv=8i+jandw=6i+j
,
findv+w.
A) 10 2 B) 2 C) 102 D) 7 5
Findthequantityifv=5i–7jandw=3i+2j.
8) v+w
A) 74+13 B) 87 C) 2 6+13 D) 39
9) v–w
A) 74–13 B) 85 C) 2 6–13 D) 11
10) v+w
A) 89 B) 74+13 C) 39 D) 2 6+13
11) v–w
A) 85 B) 74–13 C) 77 D) 2 6–13
5 FindaUnitVector
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheunitvectorhavingthesamedirectionasv.
1) v=7i
A) u=iB) u=7iC) u=49iD) u=1
7i
2) v=–2j
A) u=–jB) u=–2jC) u=4jD) u=–1
2j
3) v=3i+4j
A) u=3
5i+4
5jB) u=15i+20jC) u=–4
5i–3
5jD) u=5
3i+5
4j
4) v=12i+5j
A) u=12
13i+5
13jB) u=156i+65jC) u=–5
13i–12
13jD) u=13
12i+13
5j
5) v=–3i+j
A) u=–310
10 i+10
10 jB) u=–310
i+10j
C) u=–310
10 i–10
10 jD) u=–10
3i+10j
6 FindaVectorfromItsDirectionandMagnitude
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Findavectorvwhosemagnitudeis20 andwhosecomponentintheidirectionisfivetimes the
componentinthejdirection.
A) v=50
13 26i+10
13 26jor v=–50
13 26i–10
13 26j
B) v=10
13 26i+50
13 26jor v=–10
13 26i–50
13 26j
C) v=–50
13 26i+10
13 26jor v=50
13 26i–10
13 26j
D) v=10
13 26i–50
13 26jor v=–10
13 26i+50
13 26j
2) IfP=(6,–6)andQ=(x,30),findallnumbersxsuchthatthevectorrepresentedbyPQ haslength60.
A) {–42
,
54} B) {–54
,
54} C) {–54
,
–42} D) {–42
,
60}
Writethevectorvintheformai+bj,givenitsmagnitudevandtheangle
α
itmakeswiththepositivex–axis.
3) v=5
,
α=120°
A) v=5–5
2i+53
2jB) v=553
2i–5
2jC) v=5–2
2i+2
2jD) v=55
2i–53
2j
4) v=11
,
α=225°
A) v=11 –2
2i–2
2jB) v=11 –11 3
2i–11
2j
C) v=11 –11
2i–11 3
2jD) v=11 2
2i+2
2j
5) v=15
,
α=30°
A) v=15 15 3
2i+1
2jB) v=15 15
2i+15 3
2j
C) v=15 2
2i+2
2jD) v=15 –15 3
2i+15
2j
6) v=8
,
α=90°
A) v=8jB) v=8iC) v=8i–8j)D)
v=8i+8j
7) v=12
,
α=0°
A) v=12iB) v=12jC) v= –12jD) v=12i+12j
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
8) Atruckpushesaloadof45tonsupahillwithaninclinationof35°.ExpresstheforcevectorFintermsofi
andj.RoundthecomponentsofFtotwodecimalplaces.
9) Twoforces,F1ofmagnitude35newtons(N)andF2ofmagnitude55newtons,actonanobjectatanglesof
45°and–60°(respectively)withthepositivex–axis.Findthedirectionandmagnitudeoftheresultant
force;thatis,findF1+F2.Roundthedirectionandmagnitudetotwodecimalplaces.
10) Twoforces,F1ofmagnitude60newtons(N)andF2ofmagnitude70newtons,actonanobjectatanglesof
40°and130°(respectively)withthepositivex–axis.Findthedirectionandmagnitudeoftheresultant
force;thatis,findF1+F2.Roundthedirectionandmagnitudetotwodecimalplaces.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
11) Twoforcesofmagnitude25poundsand40poundsactonanobject.Theforceof40lbactsalongthe
positivex–axis,andtheforceof25lbactsatanangleof80°withthepositivex–axis.Findthedirectionand
magnitudeoftheresultantforce.Roundthedirectionandmagnitudetothenearestwholenumber.
A) Direction:29°;magnitude:51lb B) Direction:40°;magnitude:47lb
C) Direction:51°;magnitude:51lb D) Direction:4°;magnitude:65lb
7 ModelwithVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Anaudiospeakerthatweighs50poundshangsfromtheceilingofarestaurantfromtwocablesasshown
inthefigure.Totwodecimalplaces,whatisthetensioninthetwocables?
A) Tensioninrightcable:35.90lb;tensioninleftcable:41.59lb
B) Tensioninrightcable:35.90lb;tensioninleftcable:14.10lb
C) Tensioninrightcable:41.59lb;tensioninleftcable:35.90lb
D) Tensioninrightcable:14.10lb;tensioninleftcable:41.59lb
2) Aboxofsuppliesthatweighs1500kilogramsissuspendedbytwocablesasshowninthefigure.Totwo
decimalplaces,whatisthetensioninthetwocables?
A) Tensioninrightcable:1098.08kg;tensioninleftcable:776.46kg
B) Tensioninrightcable:723.54kg;tensioninleftcable:776.46kg
C) Tensioninrightcable:776.46kg;tensioninleftcable:1098.08kg
D) Tensioninrightcable:776.46kg;tensioninleftcable:723.54kg
3) Atightropewalkerlocatedatacertainpointdeflectstheropeasindicatedinthefigure.Iftheweightofthe
tightropewalkeris140pounds,howmuchtensionisineachpartoftherope?Roundyouranswerstothe
nearesttenth.
4.3° 3.3°
140pounds
A) tensionintheleftpart:–365.8lb;
tensionintherightpart:–360.7lb
B) tensionintheleftpart:–210.0lb;
tensionintherightpart:136.0lb
C) tensionintheleftpart:–1954.8lb;
tensionintherightpart:–1927.8lb
D) tensionintheleftpart:–160.7lb;
tensionintherightpart:104.1lb
4) Atastatefairtruckpull,twopickuptrucksareattachedtothebackendofamonstertruckasillustratedin
thefigure.Oneofthepickupspullswithaforceof1600poundsandtheotherpullswithaforceof3600
poundswithanangleof45°betweenthem.Withhowmuchforcemustthemonstertruckpullinorderto
remainunmoved?HINT:Findtheresultantforceofthetwotrucks.Roundyouranswertothenearest
tenth.
1600lb
45°
3600lb
A) Thetruckmustpullwithaforceof4864.8 lb. B) Thetruckmustpullwithaforceof2715.5 lb.
C) Thetruckmustpullwithaforceof4594.1 lb. D) Thetruckmustpullwithaforceof2194.1 lb.
10.5 TheDotProduct
1 FindtheDotProductofTwoVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthedotproductv·w.
1) v=–i–j
,
w=–i–j
A) 2 B) 0 C) 1 D) –1
2) v=–3i
,
w=–4j
A) 0 B) 12 C) 5 D) –7
3) v=i+2j, w=5i–j
A) 3 B) 7 C) –3D)0
4) v=6i–3j, w=8i+j
A) 45 B) 51 C) –18 D) 0
5) v=–15i+7j
,
w=12i–4j
A) –208 B) –152 C) –180 D) –28
6) v=13i+11j
,
w=–9i–10j
A) –227 B) –7C)
–117 D) –110
2 FindtheAnglebetweenTwoVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheanglebetweenvandw.Roundyouranswertoonedecimalplace,ifnecessary.
1) v=2i–2j, w=3i+2j
A) 78.7° B) 88.7° C) 39.4° D) 29.4°
2) v=–5i+7j
,
w=–6i–4j
A) 88.2° B) 90.9° C) 20.7° D) 110.8°
3) v=5i
,
w=j
A) 90° B) 180° C) 0° D) 270°
3 DetermineWhetherTwoVectorsAreParallel
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Whichofthefollowingvectorsisparalleltov=–10i–8j?
A) w=20i+16jB) w=3i–5jC) w=4i+4jD) w= –20i+25j
2) Whichofthefollowingvectorsisparalleltov=9i+5j?
A) w=3
2i+5
6jB) w=45i–25jC) w=–15
2i+25
6jD) w=15
2i–25
6j
4 DetermineWhetherTwoVectorsAreOrthogonal
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Whichofthefollowingvectorsisorthogonalto20i–8j?
A) w=–10i–25jB) w=20i+4jC) w=15i–6jD) w=4i+3j
Statewhetherthevectorsareparallel,orthogonal,orneither.
2) v=3i+j
,
w=i–3j
A) Orthogonal B) Parallel C) Neither
3) v=4i+3j
,
w=3i–4j
A) Orthogonal B) Parallel C) Neither
4) v=4i–j
,
w=8i–2j
A) Parallel B) Orthogonal C) Neither
5) v=4i+3j
,
w=8i+6j
A) Parallel B) Orthogonal C) Neither
6) v=4i–2j
,
w=4i+2j
A) Parallel B) Orthogonal C) Neither
7) v=i+3j,w=i–2j
A) Orthogonal B) Parallel C) Neither
Solvetheproblem.
8) Findasothatthevectorsv=i–ajandw=3i+2jareorthogonal.
A) 3
2B) –3
2C) –2
3D) 2
3
5 DecomposeaVectorintoTwoOrthogonalVectors
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Decomposevintotwovectorsv1andv2,wherev1isparalleltowandv2isorthogonaltow.
1) v=i+7j
,
w=i+j
A) v1=4i+4j),v2=–3i+3jB) v1=9
2i+9
2j,v2=–7
2i+5
2j
C) v1=4i+4j
,
v2=3i–3jD) v1=8i+8j
,
v2= –6i+6j
2) v=i–4j
,
w=2i+j
A) v1=–4
5i–2
5j,v2=9
5i–18
5jB) v1=–4
5i–2
5j,v2=7
5i–18
5j
C) v1=–1i–1
2j,v2=2i–7
2jD) v1=–4
5i–2
5j,v2=–7
5i–26
5j
3) v=–2i–5j
,
w=–3i+j
A) v1=–3
10i+1
10j,v2=–17
10i–51
10jB) v1=–3
10i+1
10j,v2=–21
10i–21
5j
C) v1=–1
3i+1
9j,v2=–5
3i–46
9jD) v1=–3
10i+1
10j,v2=13
10i–61
10j
4) v=–3i–2j
,
w=–3i–j
A) v1=–33
10i+11
10j,v2=3
10i–31
10jB) v1=–33
10i+11
10j,v2=–41
10i–31
10j
C) v1=–11
3i+11
9j,v2=2
3i–29
9jD) v1=–33
10i+11
10j,v2=–9
10i–27
10j
Solvetheproblem.
5) AnSUVweighing5600poundsisparkedonastreetwhichhasaninclineof8°.Findtheforcerequiredto
keeptheSUVfromrollingdownthehillandtheforceoftheSUVperpendiculartothehill.Roundthe
forcestothenearesthundredth.
A) 779.37lband5545.50lb B) 1164.31 lband5477.63lb
C) 390.64lband5586.36lb D) 389.68 lband2772.75lb
6 ComputeWork
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.Roundyouranswertothenearesttenth.
1) Findtheworkdonebyaforceof8poundsactinginthedirectionof39° tothehorizontalinmovingan
object9feetfrom(0,0)to(9,0).
A) 56.0ft–lb B) 45.3 ft–lb C) 111.9 ft–lb D) 59.7 ft–lb
2) Awagonispulledhorizontallybyexertingaforceof60poundsonthehandleatanangleof25° tothe
horizontal.Howmuchworkisdoneinmovingthewagon50feet?ʺ.
A) 2718.9ft–lb B) 1267.9ft–lb C) 1617.4ft–lb D) 2110.8ft–lb
3) Apersonispullingafreightcartwithaforceof48 pounds.Howmuchworkisdoneinmovingthecart50
feetifthecartʹshandlemakesanangleof19°withtheground?
A) 2269.2ft–lb B) 781.4 ft–lb C) 78.1 ft–lb D) 2328.7 ft–lb
4) Findtheworkdonebyaforceof200poundsactinginthedirection–i+2jinmovinganobject75feetfrom
(0,0)to(–75,0).
A) 6708.2ft–lb B) 15,000.0ft–lb C) 13,416.1ft–lb D) 8944.9ft–lb
Ch.10 PolarCoordinates;Vectors
AnswerKey
10.1 PolarCoordinates
1 PlotPointsUsingPolarCoordinates