Question 1

Find the dot product v · w.
- v=2i+j+3k\mathbf { v } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }v=2i+j+3k  and  w=i+2j−2k\mathbf { w } = \mathrm { i } + 2 \mathbf { j } - 2 \mathbf { k }w=i+2j−2k

Accepted Answer

A)    −2- 2−2 
B)    −7- 7−7 
C)   2
D)    −6- 6−6

Question 2

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.
- −6- 6−6

Accepted Answer

A)    6(cos⁡270∘+isin⁡270∘) 6 \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right) 6(cos270∘+isin270∘)  
B)    6(cos⁡180∘+isin⁡180∘) 6 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right) 6(cos180∘+isin180∘)  
C)    6(cos⁡0∘+isin⁡0∘) 6 \left( \cos 0 ^ { \circ } + \mathrm { i } \sin 0 ^ { \circ } \right) 6(cos0∘+isin0∘)  
D)    6(cos⁡90∘+isin⁡90∘) 6 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) 6(cos90∘+isin90∘)

Question 3

Choose the one alternative that best completes the statement or answers the question.
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
- (5,3π4) \left( 5 , \frac { 3 \pi } { 4 } \right) (5,43π​)

Accepted Answer

A)    (−522,522) \left( \frac { - 5 \sqrt { 2 } } { 2 } , \frac { 5 \sqrt { 2 } } { 2 } \right) (2−52c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​,252c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)  
B)    (−522,−522) \left( \frac { - 5 \sqrt { 2 } } { 2 } , \frac { - 5 \sqrt { 2 } } { 2 } \right) (2−52c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​,2−52c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)  
C)    (522,−522) \left( \frac { 5 \sqrt { 2 } } { 2 } , \frac { - 5 \sqrt { 2 } } { 2 } \right) (252c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​,2−52c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)  
D)    (522,522) \left( \frac { 5 \sqrt { 2 } } { 2 } , \frac { 5 \sqrt { 2 } } { 2 } \right) (252c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​,252c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)

Question 4

Find the distance from P1 to P2.
- P1=(0,0,0) \mathrm { P } _ { 1 } = ( 0,0,0 ) P1​=(0,0,0)   and  P2=(2,4,3) \mathrm { P } _ { 2 } = ( 2,4,3 ) P2​=(2,4,3)

Accepted Answer

A)    23\sqrt { 23 }23c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
B)    17\sqrt { 17 }17c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
C)    29\sqrt { 29 }29c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)    333 \sqrt { 3 }33c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​

Question 5

Find the angle between v and w. Round your answer to one decimal place, if necessary.
- v=−5i+7j,w=−6i−4jv = - 5 i + 7 \mathbf { j } , \quad \mathbf { w } = - 6 \mathbf { i } - 4 \mathbf { j }v=−5i+7j,w=−6i−4j

Accepted Answer

A)    110.8∘110.8 ^ { \circ }110.8∘ 
B)    90.9∘90.9 ^ { \circ }90.9∘ 
C)    20.7∘20.7 ^ { \circ }20.7∘ 
D)    88.2∘88.2 ^ { \circ }88.2∘

Question 6

Find the area of the parallelogram.
- P1(0,0,0) ,P2(4,2,1) ,P3(−2,3,1) \mathrm { P } _ { 1 } ( 0,0,0 )  , \mathrm { P } _ { 2 } ( 4,2,1 )  , \mathrm { P } _ { 3 } ( - 2,3,1 ) P1​(0,0,0) ,P2​(4,2,1) ,P3​(−2,3,1)

Accepted Answer

A)    293\sqrt { 293 }293c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
B)    21\sqrt { 21 }21c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
C)    233\sqrt { 233 }233c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)    14\sqrt { 14 }14c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​

Question 7

Solve the problem. Round your answer to the nearest tenth.
-A person is pulling a freight cart with a force of 40 pounds. How much work is done in moving the cart 30 feet if the cart's handle makes an angle of 22° with the ground?

Accepted Answer

A)   1,112.6 ft-lb
B)   45.0 ft-lb
C)   449.5 ft-lb
D)   1,147.6 ft-lb

Question 8

Graph the polar equation.
- r=csc⁡θ−2,0<θ<πr=\csc 	heta-2,0<	heta<\pir=cscθ−2,0<θ<π

Accepted Answer

A)   
B)  
C)  
D)

Question 9

The rectangular coordinates of a point are given. Find polar coordinates for the point.
- (100,−30) ( 100 , - 30 ) (100,−30)   Round the polar coordinates to two decimal places, with  θ	hetaθ  in degrees.

Accepted Answer

A)    (104.40,16.70∘) \left( 104.40,16.70 ^ { \circ } \right) (104.40,16.70∘)  
B)    (104.40,−106.70∘) \left( 104.40 , - 106.70 ^ { \circ } \right) (104.40,−106.70∘)  
C)    (104.40,−16.70∘) \left( 104.40 , - 16.70 ^ { \circ } \right) (104.40,−16.70∘)  
D)    (104.40,106.70∘) \left( 104.40,106.70 ^ { \circ } \right) (104.40,106.70∘)  C

Question 10

Choose the one alternative that best completes the statement or answers the question.
Find the direction angles of the vector. Round to the nearest degree, if necessary.
- v=i−j+2k\mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }v=i−j+2k

Accepted Answer

A)    α=66∘,β=114∘,γ=35∘\alpha = 66 ^ { \circ } , \beta = 114 ^ { \circ } , \gamma = 35 ^ { \circ }α=66∘,β=114∘,γ=35∘ 
B)    α=80∘,β=100∘,γ=71∘\alpha = 80 ^ { \circ } , \beta = 100 ^ { \circ } , \gamma = 71 ^ { \circ }α=80∘,β=100∘,γ=71∘ 
C)    α=63∘,β=117∘,γ=26∘\alpha = 63 ^ { \circ } , \beta = 117 ^ { \circ } , \gamma = 26 ^ { \circ }α=63∘,β=117∘,γ=26∘ 
D)    α=69∘,β=111∘,γ=45∘\alpha = 69 ^ { \circ } , \beta = 111 ^ { \circ } , \gamma = 45 ^ { \circ }α=69∘,β=111∘,γ=45∘ A

Question 11

Plot the point given in polar coordinates.
- (4,−5π4) \left( 4 , \frac { - 5 \pi } { 4 } \right) (4,4−5π​)

Accepted Answer

A)  
B)  
C)  
D)

Question 12

Find the unit vector having the same direction as v.
- v=−3i+j\mathbf { v } = - 3 \mathbf { i } + \mathbf { j }v=−3i+j

Accepted Answer

A)    u=−31010i−1010j\mathbf { u } = - \frac { 3 \sqrt { 10 } } { 10 } \mathbf { i } - \frac { \sqrt { 10 } } { 10 } \mathbf { j }u=−10310c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​i−1010c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​j 
B)    u=−103i+10j\mathbf { u } = - \frac { \sqrt { 10 } } { 3 } \mathbf { i } + \sqrt { 10 } \mathbf { j }u=−310c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​i+10c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​j 
C)    u=−310i+10j\mathbf { u } = - 3 \sqrt { 10 } \mathbf { i } + \sqrt { 10 } \mathbf { j }u=−310c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​i+10c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​j 
D)    u=−31010i+1010j\mathbf { u } = - \frac { 3 \sqrt { 10 } } { 10 } \mathbf { i } + \frac { \sqrt { 10 } } { 10 } \mathbf { j }u=−10310c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​i+1010c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​j

Question 13

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-Find a unit vector normal to the plane containing $$\mathbf { u } = - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } \text { and } \mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 14

Solve the problem. Leave your answer in polar form.
- z=1−iw=1−3i\begin{array} { l } z = 1 - i \w = 1 - \sqrt { 3 } i\end{array}z=1−iw=1−3​i​ 
Find  zw\frac { z } { w }wz​ .

Accepted Answer

A)    22(cos⁡75∘+isin⁡75∘) \frac { \sqrt { 2 } } { 2 } \left( \cos 75 ^ { \circ } + i \sin 75 ^ { \circ } \right) 22c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​(cos75∘+isin75∘)  
B)    12(cos⁡75∘+isin⁡75∘) \frac { 1 } { 2 } \left( \cos 75 ^ { \circ } + i \sin 75 ^ { \circ } \right) 21​(cos75∘+isin75∘)  
C)    22(cos⁡15∘+isin⁡15∘) \frac { \sqrt { 2 } } { 2 } \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right) 22c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​(cos15∘+isin15∘)  
D)    12(cos⁡15∘+isin⁡15∘) \frac { 1 } { 2 } \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right) 21​(cos15∘+isin15∘)  C

Question 15

Write the complex number in rectangular form.
- 8(cos⁡π6+isin⁡π6) 8 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) 8(cos6π​+isin6π​)

Accepted Answer

A)    4+43i4 + 4 \sqrt { 3 } \mathrm { i }4+43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​i 
B)    14+34i\frac { 1 } { 4 } + \frac { \sqrt { 3 } } { 4 } \mathrm { i }41​+43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​i 
C)    43+4i4 \sqrt { 3 } + 4 i43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+4i 
D)    34+14i\frac { \sqrt { 3 } } { 4 } + \frac { 1 } { 4 } \mathrm { i }43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​+41​i

Question 16

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
- rsin⁡θ=4r \sin 	heta = 4rsinθ=4

Accepted Answer

A)  
B)    x=−4x = - 4x=−4 ; vertical line 4 unitsto the left of the pole
C)   x=4x = 4x=4 ; vertical line 4 unitsto the right of the pole
D)    y=−4y = - 4y=−4 ; horizontal line 4 unitsbelow the pole y=4\mathrm { y } = 4y=4 ; horizontal line 4 unitsabove the pole

Question 17

Find the indicated cross product.
- v=6i+5j−3k,w=−4i−4k\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k } , \quad \mathbf { w } = - 4 \mathbf { i } - 4 \mathbf { k }v=6i+5j−3k,w=−4i−4k 
Find  v×wv 	imes wv×w .

Accepted Answer

A)    −20i+12j−20k- 20 \mathbf { i } + 12 \mathrm { j } - 20 \mathbf { k }−20i+12j−20k 
B)    −20i+36j+20k- 20 \mathbf { i } + 36 \mathbf { j } + 20 \mathbf { k }−20i+36j+20k 
C)    −15i+34j+30k- 15 i + 34 j + 30 k−15i+34j+30k 
D)    20i+20j−36k20 \mathrm { i } + 20 \mathrm { j } - 36 \mathbf { k }20i+20j−36k

Question 18

Test the equation for symmetry with respect to the given axis, line, or pole.
- r=6+2cos⁡θ; the pole \mathrm { r } = 6 + 2 \cos 	heta 	ext {; the pole }r=6+2cosθ; the pole

Accepted Answer

A)   Symmetric with respect to the pole
B)   May or may not be symmetric with respect to the pole

Question 19

Match the point in polar coordinates with either A, B, C, or D on the graph.
- (−3,−π3) \left( - 3 , - \frac { \pi } { 3 } \right) (−3,−3π​)

Accepted Answer

A)   A
B)   B
C)   C
D)   D

Question 20

Choose the one alternative that best completes the statement or answers the question.
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
- (7,2π3) \left( 7 , \frac { 2 \pi } { 3 } \right) (7,32π​)

Accepted Answer

A)    (−72,732) \left( - \frac { 7 } { 2 } , \frac { 7 \sqrt { 3 } } { 2 } \right) (−27​,273c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)  
B)    (72,−732) \left( \frac { 7 } { 2 } , \frac { - 7 \sqrt { 3 } } { 2 } \right) (27​,2−73c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)  
C)    (−72,−732) \left( - \frac { 7 } { 2 } , \frac { - 7 \sqrt { 3 } } { 2 } \right) (−27​,2−73c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)  
D)    (72,732) \left( \frac { 7 } { 2 } , \frac { 7 \sqrt { 3 } } { 2 } \right) (27​,273c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​)

Exam 8: Polar Coordinates; Vectors

Find the dot product v · w. - $\mathbf { v } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }$ and $\mathbf { w } = \mathrm { i } + 2 \mathbf { j } - 2 \mathbf { k }$

Correct Answer
verified

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - $- 6$

Correct Answer
verified

Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( 5 , \frac { 3 \pi } { 4 } \right)$

Correct Answer
verified

Find the distance from P1 to P2. - $\mathrm { P } _ { 1 } = ( 0,0,0 )$ and $\mathrm { P } _ { 2 } = ( 2,4,3 )$

Correct Answer
verified

Find the angle between v and w. Round your answer to one decimal place, if necessary. - $v = - 5 i + 7 \mathbf { j } , \quad \mathbf { w } = - 6 \mathbf { i } - 4 \mathbf { j }$

Correct Answer
verified

Find the area of the parallelogram. - $\mathrm { P } _ { 1 } ( 0,0,0 ) , \mathrm { P } _ { 2 } ( 4,2,1 ) , \mathrm { P } _ { 3 } ( - 2,3,1 )$

Correct Answer
verified

Solve the problem. Round your answer to the nearest tenth. -A person is pulling a freight cart with a force of 40 pounds. How much work is done in moving the cart 30 feet if the cart's handle makes an angle of 22° with the ground?

Correct Answer
verified

Graph the polar equation. - $r=\csc \theta-2,0<\theta<\pi$ $Graph the polar equation. - r=\csc \theta-2,0<\theta<\pi A) B) C) D)$

Correct Answer
verified

The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( 100 , - 30 )$ Round the polar coordinates to two decimal places, with $\theta$ in degrees.

Correct Answer
verified
C

Choose the one alternative that best completes the statement or answers the question. Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$

Correct Answer
verified
A

Plot the point given in polar coordinates. - $\left( 4 , \frac { - 5 \pi } { 4 } \right)$ $Plot the point given in polar coordinates. - \left( 4 , \frac { - 5 \pi } { 4 } \right) A) B) C) D)$

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Find the unit vector having the same direction as v. - $\mathbf { v } = - 3 \mathbf { i } + \mathbf { j }$

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Solve the problem. Leave your answer in polar form. - $\begin{array} { l } z = 1 - i \\w = 1 - \sqrt { 3 } i\end{array}$ Find $\frac { z } { w }$ .

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Write the complex number in rectangular form. - $8 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$

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Find the indicated cross product. - $\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k } , \quad \mathbf { w } = - 4 \mathbf { i } - 4 \mathbf { k }$ Find $v \times w$ .

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Test the equation for symmetry with respect to the given axis, line, or pole. - $\mathrm { r } = 6 + 2 \cos \theta \text {; the pole }$

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Match the point in polar coordinates with either A, B, C, or D on the graph. - $\left( - 3 , - \frac { \pi } { 3 } \right)$ $Match the point in polar coordinates with either A, B, C, or D on the graph. - \left( - 3 , - \frac { \pi } { 3 } \right) A) A B) B C) C D) D$

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Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( 7 , \frac { 2 \pi } { 3 } \right)$

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Exam 8: Polar Coordinates; Vectors

Find the dot product v · w. - v=2i+j+3k\mathbf { v } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }v=2i+j+3k and w=i+2j−2k\mathbf { w } = \mathrm { i } + 2 \mathbf { j } - 2 \mathbf { k }w=i+2j−2k

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Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - −6- 6−6

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Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - (5,3π4) \left( 5 , \frac { 3 \pi } { 4 } \right) (5,43π​)

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Find the distance from P1 to P2. - P1=(0,0,0) \mathrm { P } _ { 1 } = ( 0,0,0 ) P1​=(0,0,0) and P2=(2,4,3) \mathrm { P } _ { 2 } = ( 2,4,3 ) P2​=(2,4,3)

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Find the angle between v and w. Round your answer to one decimal place, if necessary. - v=−5i+7j,w=−6i−4jv = - 5 i + 7 \mathbf { j } , \quad \mathbf { w } = - 6 \mathbf { i } - 4 \mathbf { j }v=−5i+7j,w=−6i−4j

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Find the area of the parallelogram. - P1(0,0,0) ,P2(4,2,1) ,P3(−2,3,1) \mathrm { P } _ { 1 } ( 0,0,0 ) , \mathrm { P } _ { 2 } ( 4,2,1 ) , \mathrm { P } _ { 3 } ( - 2,3,1 ) P1​(0,0,0) ,P2​(4,2,1) ,P3​(−2,3,1)

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Solve the problem. Round your answer to the nearest tenth. -A person is pulling a freight cart with a force of 40 pounds. How much work is done in moving the cart 30 feet if the cart's handle makes an angle of 22° with the ground?

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Graph the polar equation. - r=csc⁡θ−2,0<θ<πr=\csc \theta-2,0<\theta<\pir=cscθ−2,0<θ<π

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The rectangular coordinates of a point are given. Find polar coordinates for the point. - (100,−30) ( 100 , - 30 ) (100,−30) Round the polar coordinates to two decimal places, with θ\thetaθ in degrees.

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Choose the one alternative that best completes the statement or answers the question. Find the direction angles of the vector. Round to the nearest degree, if necessary. - v=i−j+2k\mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }v=i−j+2k

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Plot the point given in polar coordinates. - (4,−5π4) \left( 4 , \frac { - 5 \pi } { 4 } \right) (4,4−5π​)

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Find the unit vector having the same direction as v. - v=−3i+j\mathbf { v } = - 3 \mathbf { i } + \mathbf { j }v=−3i+j

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Solve the problem. Leave your answer in polar form. - z=1−iw=1−3i\begin{array} { l } z = 1 - i \\w = 1 - \sqrt { 3 } i\end{array}z=1−iw=1−3​i​ Find zw\frac { z } { w }wz​ .

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Write the complex number in rectangular form. - 8(cos⁡π6+isin⁡π6) 8 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) 8(cos6π​+isin6π​)

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Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - rsin⁡θ=4r \sin \theta = 4rsinθ=4

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Find the indicated cross product. - v=6i+5j−3k,w=−4i−4k\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k } , \quad \mathbf { w } = - 4 \mathbf { i } - 4 \mathbf { k }v=6i+5j−3k,w=−4i−4k Find v×wv \times wv×w .

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Test the equation for symmetry with respect to the given axis, line, or pole. - r=6+2cos⁡θ; the pole \mathrm { r } = 6 + 2 \cos \theta \text {; the pole }r=6+2cosθ; the pole

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Match the point in polar coordinates with either A, B, C, or D on the graph. - (−3,−π3) \left( - 3 , - \frac { \pi } { 3 } \right) (−3,−3π​)

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Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - (7,2π3) \left( 7 , \frac { 2 \pi } { 3 } \right) (7,32π​)

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Find the dot product v · w. - $\mathbf { v } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }$ and $\mathbf { w } = \mathrm { i } + 2 \mathbf { j } - 2 \mathbf { k }$

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Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - $- 6$

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Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( 5 , \frac { 3 \pi } { 4 } \right)$

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Find the distance from P1 to P2. - $\mathrm { P } _ { 1 } = ( 0,0,0 )$ and $\mathrm { P } _ { 2 } = ( 2,4,3 )$

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Find the angle between v and w. Round your answer to one decimal place, if necessary. - $v = - 5 i + 7 \mathbf { j } , \quad \mathbf { w } = - 6 \mathbf { i } - 4 \mathbf { j }$

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Find the area of the parallelogram. - $\mathrm { P } _ { 1 } ( 0,0,0 ) , \mathrm { P } _ { 2 } ( 4,2,1 ) , \mathrm { P } _ { 3 } ( - 2,3,1 )$

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Graph the polar equation. - $r=\csc \theta-2,0<\theta<\pi$ $Graph the polar equation. - r=\csc \theta-2,0<\theta<\pi A) B) C) D)$

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The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( 100 , - 30 )$ Round the polar coordinates to two decimal places, with $\theta$ in degrees.

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C

Choose the one alternative that best completes the statement or answers the question. Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$

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A

Plot the point given in polar coordinates. - $\left( 4 , \frac { - 5 \pi } { 4 } \right)$ $Plot the point given in polar coordinates. - \left( 4 , \frac { - 5 \pi } { 4 } \right) A) B) C) D)$

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Find the unit vector having the same direction as v. - $\mathbf { v } = - 3 \mathbf { i } + \mathbf { j }$

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Solve the problem. Leave your answer in polar form. - $\begin{array} { l } z = 1 - i \\w = 1 - \sqrt { 3 } i\end{array}$ Find $\frac { z } { w }$ .

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C

Write the complex number in rectangular form. - $8 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$

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Find the indicated cross product. - $\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k } , \quad \mathbf { w } = - 4 \mathbf { i } - 4 \mathbf { k }$ Find $v \times w$ .

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Test the equation for symmetry with respect to the given axis, line, or pole. - $\mathrm { r } = 6 + 2 \cos \theta \text {; the pole }$

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Match the point in polar coordinates with either A, B, C, or D on the graph. - $\left( - 3 , - \frac { \pi } { 3 } \right)$ $Match the point in polar coordinates with either A, B, C, or D on the graph. - \left( - 3 , - \frac { \pi } { 3 } \right) A) A B) B C) C D) D$

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Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( 7 , \frac { 2 \pi } { 3 } \right)$

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