Deck 39:Vectors and Dot Products

Question 1

Use the vectors $\mathbf { u } = \langle 3,5 \rangle$ , $\mathbf { v } = \langle - 2,4 \rangle$ ,and $\mathbf { w } = \langle 3 , - 4 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $\left( \mathbf { v } ^ \mathbf{ . } \mathbf { u } \right) \mathbf { w }$ ​

A) $\langle 42 , - 60 \rangle$ ;vector
B) $\langle 42 , - 56 \rangle$ ;vector
C)-58;scalar
D)-56;scalar
E) $\langle 42 , - 52 \rangle$ ;vector

Answer 1

$\langle 42 , - 56 \rangle$ ;vector

Question 2

Use the vectors $\mathbf { u } = \langle 2,2 \rangle$ , $\mathbf { v } = \langle - 2,2 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $3 \mathbf { u } \cdot \mathbf { v }$ ​

A) $\langle 0,2 \rangle$ ;vector
B)-2;scalar
C)2;scalar
D) $\langle 4 , - 4 \rangle$ ;vector
E)0;scalar

Answer 2

0;scalar

Question 3

Find the dot product of u and v.​ $$\begin{array} { l } \mathbf { u } = 6 \mathbf { i } - 4 \mathbf { j } \\\mathbf { v } = \mathbf { i } - \mathbf { j }\end{array}$$ ​

A)14
B)6
C)12
D)10
E)8

Answer 3

10

Question 4

Find the dot product of u and v.​ $$\begin{array} { l } \mathbf { u } = 6 \mathbf { i } + 4 \mathbf { j } \\\mathbf { v } = 8 \mathbf { i } - 2 \mathbf { j }\end{array}$$ ​

A)36
B)40
C)44
D)42
E)38

Question 5

Use the dot product to find the magnitude of u.​ $\mathbf { u } = 35 \mathbf { i } + 40 \mathbf { j }$ ​

A)40
B) $\sqrt { 2829 }$
C) $\sqrt { 2823 }$
D) $5 \sqrt { 113 }$
E)35

Question 6

Find the dot product of u and v.​ $$\begin{array} { l } \mathbf { u } = \langle 5,1 \rangle \\\mathbf { v } = \langle - 2,4 \rangle\end{array}$$ ​

A)-10
B)-8
C)-4
D)-6
E)-2

Question 7

Use the dot product to find the magnitude of u. ​​ $\mathbf { u } = \langle - 7,12 \rangle$ ​

A) $3 \sqrt { 21 }$
B) $\sqrt { 195 }$
C) $\sqrt { 197 }$
D) $\sqrt { 193 }$
E) $\sqrt { 191 }$

Question 8

Use the vectors $\mathbf { u } = \langle 2,4 \rangle$ , $\mathbf { v } = \langle - 4,5 \rangle$ ,and $\mathbf { w } = \langle 5 , - 4 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $( \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { w }$ ​

A)-96;scalar
B) $\langle 120 , - 92 \rangle$ ;vector
C) $\langle 120 , - 94 \rangle$ ;vector
D) $\langle 120 , - 100 \rangle$ ;vector
E) $\langle 120 , - 96 \rangle$ ;vector

Question 9

Find the dot product of u and v.​ $$\begin{array} { l } \mathbf { u } = \langle 6,14 \rangle \\\mathbf { v } = \langle - 2,2 \rangle\end{array}$$ ​

A)18
B)14
C)20
D)12
E)16

Question 10

Use the dot product to find the magnitude of u.​ $\mathbf { u } = \langle 2 , - 9 \rangle$ ​

A) $\sqrt { 89 }$
B) $\sqrt { 83 }$
C) $\sqrt { 85 }$
D) $9$
E) $\sqrt { 87 }$

Question 11

Use the dot product to find the magnitude of u.​ $\mathbf { u } = 4 \mathbf { j }$ ​

A)5
B)6
C)4
D)7
E)8

Question 12

Use the vectors $\mathbf { u } = \langle 3,5 \rangle$ , $\mathbf { v } = \langle - 2,2 \rangle$ ,and $\mathbf { w } = \langle 5 , - 3 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $( 3 \mathbf { w } \cdot \mathbf { v } ) \mathbf { u }$ ​

A) $\langle - 144 , - 236 \rangle$ ;vector
B) $\langle - 144 , - 242 \rangle$ ;vector
C) $\langle - 144 , - 240 \rangle$ ;vector
D)-240;scalar
E) $\langle - 144 , - 244 \rangle$ ;vector

Question 13

Use the vectors $$\mathbf { u } = \langle 4,5 \rangle$$ , $\mathbf { v } = \langle - 6,5 \rangle$ ,and $\mathbf { w } = \langle 6 , - 4 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $\langle \mathbf { u } \cdot \mathbf { v } \rangle - \langle \mathbf { u } \cdot \mathbf { w } \rangle$ ​

A)-1;scalar
B) $\langle 1 , - 7 \rangle$ ;vector
C) $\langle - 3 , - 3 \rangle$ ;vector
D)-3;scalar
E)-5;scalar

Question 14

Use the vectors $\mathbf { u } = \langle 2,4 \rangle$ , $\mathbf { v } = \langle - 5,3 \rangle$ and $\mathbf { w } = \langle 3 , - 1 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $\left( \mathbf { v } ^ { . } \mathbf { u } \right) - \left( \mathbf { w } ^ { . } \mathbf { v } \right)$ ​

A)16;scalar
B) $\langle 24,16 \rangle$ ;vector
C) $$\langle 0,20 \rangle$$ ;vector
D)22;scalar
E)20;scalar

Question 15

Find the angle $\theta$ between the vectors.​ $\begin{array} { l } \mathbf { u } = \langle 4,0 \rangle \\\mathbf { v } = \langle 0 , - 3 \rangle\end{array}$ ​ (Round the answer to 1 decimal place. )
​

A) $30 ^ { \circ }$
B) $110 ^ { \circ }$
C) $70 ^ { \circ }$
D) $105 ^ { \circ }$
E) $90 ^ { \circ }$

Question 16

Find the dot product of u and v.​ $$\begin{array} { l } \mathbf { u } = \langle - 7,1 \rangle \\\mathbf { v } = \langle 4 , - 5 \rangle\end{array}$$ ​

A)-31
B)-29
C)-33
D)-37
E)-35

Question 17

Find the dot product of u and v.​ $$\begin{array} { l } \mathbf { u } = \langle - 4,9 \rangle \\\mathbf { v } = \langle - 2 , - 5 \rangle\end{array}$$ ​

A)-41
B)-33
C)-37
D)-35
E)-39

Question 18

Use the dot product to find the magnitude of u.​ $\mathbf { u } = 24 \mathbf { i } - 28 \mathbf { j }$ ​

A) $4 \sqrt { 85 }$
B) $\sqrt { 1358 }$
C) $2 \sqrt { 339 }$
D) $\sqrt { 1362 }$
E) $2 \sqrt { 341 }$

Question 19

Use the vectors $\mathbf { u } = \langle 3,6 \rangle$ , $\mathbf { v } = \langle - 6,5 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $( \mathbf { u } \cdot \mathbf { v } ) \mathbf { v }$ ​

A) $\langle - 72,60 \rangle$ ;vector
B) $\langle - 72,64 \rangle$ ;vector
C)-72;scalar
D)60;scalar
E) $\langle - 72,58 \rangle$ ;vector

Question 20

Use the vector $\mathbf { u } = \langle 3,2 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.​ $\mathbf { u } ^\mathbf{.}\mathbf { u } ^ { }$ ​

A) $\langle 13,15 \rangle$ ;vector
B)15;scalar
C)13;scalar
D)11;scalar
E) $\langle 17,9 \rangle$ ;vector

Question 21

Given vectors $\mathbf { u } = \left\langle u _ { 1 } , u _ { 2 } \right\rangle$ , $\mathbf { v } = \left\langle v _ { 1 } , v _ { 2 } \right\rangle$ and $\mathbf { w } = \left\langle w _ { 1 } , w _ { 2 } \right\rangle$ determine whether the result of the following expression is a vector or a scalar. $6 w \cdot 3 w$ ​

A)vector
B)scalar

Question 22

Given vectors $\mathbf { u } = \langle - 2 , - 3 \rangle$ and $v = \langle 1,3 \rangle$ determine the quantity indicated below. $\mathbf { u } \cdot 2 \mathbf { v }$

A)18
B)-20
C)-22
D)0
E)-6

Question 23

Given $\mathbf { u } = \langle 3 , - 2 \rangle$ and $\mathbf { v } = \langle 1 , - 3 \rangle$ ,find $u\cdot v$ .

A)3
B)-3
C)-7
D)9
E)-11

Question 24

Find the angle θ between the vectors. ​​ $\begin{array} { l } \mathbf { u } = - 7 \mathbf { i } - 8 \mathbf { j } \\\mathbf { v } = - 9 \mathbf { i } + 2 \mathbf { j }\end{array}$ ​
(Round the answer to 2 decimal places. )
​

A) $71.34 ^ { \circ }$
B) $66.34 ^ { \circ }$
C) $61.34 ^ { \circ }$
D) $81.34 ^ { \circ }$
E) $76.34 ^ { \circ }$

Question 25

Find the angle between the vectors u and v if $\mathbf { u } = \langle 3 , - 3 \rangle$ ,and $\mathbf { v } = \langle 1 , - 1 \rangle$ Round your answer to two decimal places.

A)1.41°
B)0.00°
C)-1.24°
D)-1.93°
E)0.96°

Question 26

Use the dot product to find the magnitude of u if $\mathbf { u } = \langle 6 , - 2 \rangle$

A) $\| \mathbf { u } \| = 4$
B) $\| \mathbf { u } \| = 12$
C) $\| \mathbf { u } \| = 20$
D) $\| \mathbf { u } \| =$ $4 \sqrt { 10 }$
E)​ $\| \mathbf { u } \| =$ $2 \sqrt { 10 }$

Question 27

Find the projection of u onto v.​ $\begin{array} { l } \mathbf { u } = \langle 3,6 \rangle \\\mathbf { v } = \langle 4,6 \rangle\end{array}$ ​

A) $13 \langle 72,48 \rangle$
B) $\frac { 1 } { 13 } \langle 3,4 \rangle$
C) $\frac { 13 } { 12 } \langle 48,72 \rangle$
D) $\frac { 1 } { 13 } \langle 48,72 \rangle$
E) $\frac { 12 } { 13 } \langle 48,72 \rangle$

Question 28

Given vectors $\mathbf { u } = \langle - 1 , - 2 \rangle$ and $\mathbf { v } = \langle 3,5 \rangle$ determine the quantity indicated below. $( 4 \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { u }$

A) $\langle 32,64 \rangle$
B) $\langle 104,208 \rangle$
C) $\langle - 8 , - 16 \rangle$
D) $\langle 8,16 \rangle$
E) $\langle - 64 , - 128 \rangle$

Question 29

A force of $y = 50$ pounds exerted at an angle of $30 ^ { \circ }$ above the horizontal is required to slide a table across a floor (see figure).The table is dragged $x = 12$ feet.Determine the work done in sliding the table.​ ​

A)539.6 ft-lb
B)599.6 ft-lb
C)519.6 ft-lb
D)579.6 ft-lb
E)559.6 ft-lb

Question 30

Determine the work done by a crane lifting a 2,800-pound car 7 feet.

A)19,600 ft-lb
B)19,620 ft-lb
C)19,610 ft-lb
D)19,640 ft-lb
E)19,630 ft-lb

Question 31

Find the projection of u onto v.​ $\begin{array} { l } \mathbf { u } = \langle - 5 , - 5 \rangle \\\mathbf { v } = \langle - 5 , - 1 \rangle\end{array}$ ​

A) $\frac { 1 } { 13 } \langle - 75 , - 15 \rangle$
B) $13 \langle - 15 , - 75 \rangle$
C) $\frac { 1 } { 13 } \langle - 5 , - 5 \rangle$
D) $\frac { 13 } { 15 } \langle - 75 , - 15 \rangle$
E) $\frac { 15 } { 13 } \langle - 75 , - 15 \rangle$

Question 32

Determine the work done by a person lifting a 240-newton bag of sugar 2 meters.

A)510 N-m
B)480 N-m
C)520 N-m
D)500 N-m
E)490 N-m

Question 33

Find the angle between the vectors u and v if $\mathbf { u } = 3 \mathbf { i } - 2 \mathbf { j }$ and $\mathbf { u } = 3 \mathbf { i } + 2 \mathbf { j }$ Round your answer to two decimal places.

A)66.14°
B)68.34°
C)67.38°
D)65.45°
E)68.79°

Question 34

A tractor pulls a log 900 meters,and the tension in the cable connecting the tractor and log is approximately 16,691 newtons.The direction of the force is $35 ^ { \circ }$ above the horizontal.Approximate the work done in pulling the log. ​
(Round the answer to two decimal places. )
​

A)12,305,260.09 N-m
B)12,305,220.09 N-m
C)12,305,300.09 N-m
D)12,305,280.090 N-m
E)12,305,240.09 N-m

Question 35

Use the dot product to find the magnitude of u if $\mathbf { u } = - 3 \mathbf { i } + 6 \mathbf { j }$

A) $\| \mathbf { u } \| = 18$
B) $\| \mathbf { u } \| =$ $6 \sqrt { 5 }$
C) $\| \mathbf { u } \| = 18$ .
D)​ $\| \mathbf { u } \| =$ $3 \sqrt { 5 }$
E) $\| \mathbf { u } \| = 23$

Question 36

Find the angle θ between the vectors.​ $\begin{array} { l } \mathbf { u } = \langle 6,2 \rangle \\\mathbf { v } = \langle 7,0 \rangle\end{array}$ ​ (Round the answer to 2 decimal places. )
​

A) $18.43 ^ { \circ }$
B) $33.43 ^ { \circ }$
C) $23.43 ^ { \circ }$
D) $28.43 ^ { \circ }$
E) $38.43 ^ { \circ }$

Question 37

Find the projection of u onto v.​ $\begin{array} { l } \mathbf { u } = \langle 0,4 \rangle \\\mathbf { v } = \langle 3,20 \rangle\end{array}$ ​

A) $\frac { 1 } { 409 } \langle 0,3 \rangle$
B) $409 \langle 1600,240 \rangle$
C) $\frac { 80 } { 409 } \langle 240,1600 \rangle$
D) $\frac { 409 } { 80 } \langle 240,1600 \rangle$
E) $\frac { 1 } { 409 } \langle 240,1600 \rangle$

Question 38

Given $\mathbf { u } = - 3 \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { u } = 4 \mathbf { i } - \mathbf { j }$ ,find $u\cdot v$ .

A)-10
B)-14
C)11
D)-5
E)-12

Question 39

Find the angle θ between the vectors. ​​ $\begin{array} { l } \mathbf { u } = 6 \mathbf { i } - 1 \mathbf { j } \\\mathbf { v } = 7 \mathbf { i } + 2 \mathbf { j }\end{array}$ ​
(Round the answer to 2 decimal places. )
​

A) $25.41 ^ { \circ }$
B) $45.41 ^ { \circ }$
C) $30.41 ^ { \circ }$
D) $35.41 ^ { \circ }$
E) $40.41 ^ { \circ }$

Question 40

Find the projection of u onto v.Then write u as the sum of two orthogonal vectors,one of which is $\operatorname { proj } _ { \mathrm { v } } \mathbf { u }$ .​ ​​ $\mathbf { u } = \langle 30,5 \rangle$ $\mathbf { v } = \langle 1 , - 6 \rangle$ ​

A)​ $\langle 0,0 \rangle , \langle - 30,5 \rangle$
B)​ $\langle 0,0 \rangle , \langle 30,5 \rangle$
C)​ $\langle 0,0 \rangle , \langle - 1 , - 6 \rangle$
D)​ $\langle 0,0 \rangle , \langle 30 , - 5 \rangle$
E)​ $\langle 0,0 \rangle , \langle 1 , - 6 \rangle$

Question 41

A 475-pound trailer is sitting on an exit ramp inclined at 32° on Highway 35.How much force is required to keep the trailer from rolling back down the exit ramp? Round your answer to two decimal places.

A)383.93 pounds
B)365.05 pounds
C)402.82 pounds
D)327.27 pounds
E)251.71 pounds

Question 42

Determine whether u are v and orthogonal,parallel,or neither. $$\mathbf { u } = \langle - 1,5 \rangle , \mathbf { v } = \langle - 10 , - 3 \rangle$$

A)neither
B)orthogonal
C)parallel

Question 43

Given $\mathbf { u } = - \mathbf { i } + 3 \mathbf { j }$ and $\mathbf { v } = 5 \mathbf { i } + 4 \mathbf { j }$ ,find $$u^.v$$

A)7
B)-5
C)-17
D)11
E)-19

Question 44

A force of 45 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal.The crate is moved 31 feet horizontally.How much work has been accomplished? Round your answer to one decimal place.

A)1,301.6 foot-pounds
B)1,395.0 foot-pounds
C)1,610.8 foot-pounds
D)1,208.1 foot-pounds
E)1,456.2 foot-pounds

Question 45

Determine whether u are v and orthogonal,parallel,or neither. $$\mathbf { u } = \left\langle \frac { - 4 } { 3 } , \frac { 3 } { 2 } \right\rangle , \mathbf { v } = \langle 16 , - 18 \rangle$$

A)neither
B)parallel
C)orthogonal

Question 46

Find the angle between the vectors u and v. $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $v = \cos \left( \frac { 3 \pi } { 4 } \right) i + \sin \left( \frac { 3 \pi } { 4 } \right) j$ ​

A) $40 ^ { \circ }$
B) $15 ^ { \circ }$
C) $75 ^ { \circ }$
D) $105 ^ { \circ }$
E) $70 ^ { \circ }$

Question 47

Determine $$u^.v$$ if $\| \mathbf { u } \| = 5 , \| \mathbf { v } \| = 6$ ,and $\theta = \frac { \pi } { 3 }$ where θ is the angle between u and v.Round answer to two decimal places.

A)21.21
B)20.49
C)15.00
D)18.11
E)25.98

Question 48

Use vectors to find the measure of the angle at vertex B of triangle ABC,when $A = ( 5,4 )$ , $B = ( - 2,4 )$ ,and $C = ( - 3 , - 4 )$ .Round answer to two decimal places.

A)95.94°
B)98.47°
C)97.13°
D)95.04°
E)99.65°

Question 49

Use the dot product to find the magnitude of u if $\mathbf { u } = - 5 \mathbf { i } - 2 \mathbf { j }$

A) $\| \mathbf { u } \| =$ 10
B) $\| \mathbf { u } \| =$ $2 \sqrt { 29 }$
C) $\| \mathbf { u } \| = 7$
D) $\| \mathbf { u } \| = 15$
E) $\| \mathbf { u } \| =$ $\sqrt { 29 }$

Question 50

The vector $\mathbf { u } = \langle 3900,4000 \rangle$ gives the number of units of two models of laptops produced by a company.The vector $\mathbf { u } = \langle 1450,1000 \rangle$ gives the prices (in dollars)of the two models of laptops,respectively.Identify the vector operation used to increase revenue by 6%.

A)​1.06(u + v)
B)1.06 $$\left\| \mathbf { u } ^ { . } \mathbf { v } \right\|$$
C) $\mathbf { u } ^. ( 1.06 ) \| \mathbf { v } \|$
D)1.06 $\left( \mathbf { u } ^ { . } \mathbf { v } \right)$
E)1.06 $\| \mathbf { u } \| ^ { . } \mathbf { v }$

Question 51

Use the dot product to find the magnitude of u if $\mathbf { u } = \langle 3,2 \rangle$

A) $\| \mathbf { u } \| =$ $2 \sqrt { 13 }$
B) $\| \mathbf { u } \| =$ $\sqrt { 13 }$
C) $\| \mathbf { u } \| = 6$
D) $\| \mathbf { u } \| = 5$
E) $\| \mathbf { u } \| = 7$

Question 52

Given vectors $\mathbf { u } = \langle - 1,3 \rangle$ and $\mathbf { v } = \langle 1,4 \rangle$ determine the quantity indicated below. $- 2 \mathbf { u }{ \cdot}- 4 \mathbf { v }$

A)-48
B)-56
C)-46
D)88
E)20

Question 53

The vector $\mathbf { u } = \langle 2700,4700 \rangle$ gives the number of units of two models of laptops produced by a company.The vector $\mathbf { v } = \langle 1200,900 \rangle$ gives the prices (in dollars)of the two models of laptops,respectively.Use dot products to determine the revenue for these two laptops if the price of each is increased by 6%.

A)$7,664,400
B)$7,791,300
C)$7,918,200
D)$7,723,800
E)$7,768,800

Question 54

Find the angle between the vectors u and v.​ $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $\mathbf { v } = \cos \left( \frac { 5 \pi } { 4 } \right) \mathbf { i } + \sin \left( \frac { 5 \pi } { 4 } \right) \mathbf { j }$

A) $40 ^ { \circ }$
B) $75 ^ { \circ }$
C) $105 ^ { \circ }$
D) $165 ^ { \circ }$
E) $175 ^ { \circ }$

Question 55

Find the projection of u onto v if $\mathbf { u } = \langle 5 , - 3 \rangle$ , $$\mathbf { v } = \langle - 1,4 \rangle$$ . ​

A)​ $\left\langle \frac { - 17 \sqrt { 17 } } { 17 } , \frac { 51 \sqrt { 17 } } { 17 } \right\rangle$
B)​ $\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { 68 \sqrt { 17 } } { 17 } \right\rangle$
C)​ $$\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { 17 \sqrt { 17 } } { 17 } \right\rangle$$
D)​ $$\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { - 68 \sqrt { 17 } } { 17 } \right\rangle$$
E)​ $\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { 51 \sqrt { 17 } } { 17 } \right\rangle$

Question 56

Use vectors to find the measure of the angle at vertex B of triangle ABC,when $A = ( 5,1 )$ , $B = ( - 5,1 )$ ,and $C = ( - 3 , - 5 )$ .Round answer to two decimal places.

A)69.48°
B)71.57°
C)72.91°
D)70.38°
E)74.09°

Question 57

Find the angle between the vectors u and v if $\mathbf { u } = \langle 3,4 \rangle$ ,and $\mathbf { v } = \langle - 1 , - 2 \rangle$ Round answer to two decimal places.

A)167.77°
B)168.46°
C)169.70°
D)170.66°
E)171.11°

Question 58

Find the angle between the vectors u and v if $\mathbf { u } = \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j }$ Round answer to two decimal places.

A)36.87°
B)37.83°
C)38.28°
D)34.94°
E)35.63°

Question 59

Determine $u.v$ if $\| \mathbf { u } \| = 5$ , $\| \mathbf { v } \| = 4$ ,and $\theta = \frac { \pi } { 4 }$ where θ is the angle between u and v.Round answer to two decimal places.

A)15.73
B)12.07
C)10.00
D)17.32
E)14.14

Question 60

Given $\mathbf { u } = \langle - 5 , - 6 \rangle$ and $\mathbf { v } = \langle - 6 , - 5 \rangle$ ,find $$u^.v$$ .

A)0
B)30
C)60
D)61
E)-11

Question 61

Determine whether u are v and orthogonal,parallel,or neither. $\mathbf { u } = \left\langle \frac { 1 } { 3 } , \frac { 5 } { 2 } \right\rangle$ , $\mathbf { v } = \langle - 4 , - 30 \rangle$

A)orthogonal
B)parallel
C)neither

Question 62

Determine whether u are v and orthogonal,parallel,or neither. $\mathbf { u } = \langle - 4 , - 7 \rangle , \mathbf { v } = \langle 14,24 \rangle$

A)orthogonal
B)parallel
C)neither

Question 63

Find the projection of u onto v if , $\mathbf { u } = \langle 4,2 \rangle$ , $\mathbf { v } = \langle 5 , - 2 \rangle$ . ​

A)​ $\left\langle \frac { 80 } { \sqrt { 29 } } , \frac { 32 } { \sqrt { 29 } } \right\rangle$
B)​ $\left\langle \frac { 64 } { \sqrt { 29 } } , \frac { 80 } { \sqrt { 29 } } \right\rangle$
C)​ $\left\langle \frac { 64 } { \sqrt { 29 } } , \frac { 32 } { \sqrt { 29 } } \right\rangle$
D)​ $\left\langle \frac { 64 } { \sqrt { 29 } } , - \frac { 32 } { \sqrt { 29 } } \right\rangle$
E)​ $\left\langle \frac { 80 } { \sqrt { 29 } } , - \frac { 32 } { \sqrt { 29 } } \right\rangle$

Question 64

​Find the projection of u onto v if , $\mathbf { u } = \langle - 5 , - 4 \rangle$ , $\mathbf { v } = \langle 2 , - 5 \rangle$ .

A)​ $\left\langle \frac { 20 \sqrt { 29 } } { 29 } , - \frac { 50 \sqrt { 29 } } { 29 } \right\rangle$
B)​ $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , - \frac { 50 \sqrt { 29 } } { 29 } \right\rangle$
C)​ $$\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , - \frac { 40 \sqrt { 29 } } { 29 } \right\rangle$$
D)​ $$\left\langle - \frac { 20 \sqrt { 29 } } { 29 } , - \frac { 40 \sqrt { 29 } } { 29 } \right\rangle$$
E)​ $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , \frac { 20 \sqrt { 29 } } { 29 } \right\rangle$

Question 65

The vector $\mathbf { u } = \langle 2500,3000 \rangle$ gives the number of units of two models of laptops produced by a company.The vector $\mathbf { v } = \langle 2000,1000 \rangle$ gives the prices (in dollars)of the two models of laptops,respectively.Use dot products to determine the revenue for these two laptops if the price of each is increased by 3.5%.

A)$8,186,667
B)$8,175,000
C)$8,280,000
D)$8,227,500
E)$8,105,000

Question 66

The vector $\mathbf { u } = \langle 3800,5400 \rangle$ gives the number of units of two models of laptops produced by a company.The vector $\mathbf { v } = \langle 1350,1000 \rangle$ gives the prices (in dollars)of the two models of laptops,respectively.Identify the vector operation used to increase revenue by 3.5%.

A) $\mathbf { u } \cdot ( 1.035 ) \| \mathbf { v } \|$
B) $1.035 ( \mathbf { u } \cdot \mathbf { v } )$
C) $1.035 \left\| \mathbf { u } ^ { . } \mathbf { v } \right\|$
D) $1.035 \| \mathbf { u } \| \cdot \mathbf { v }$
E) $1.035 ( \mathbf { u } + \mathbf { v } )$

Question 67

Find the projection of u onto v if $\mathbf { u } = \langle 4 , - 5 \rangle$ , $\mathbf { v } = \langle 3 , - 1 \rangle$

A)​ $\left\langle \frac { 51 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$
B)​ $$\left\langle \frac { - 51 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$$
C)​ $$\left\langle \frac { 51 } { \sqrt { 10 } } , \frac { - 51 } { \sqrt { 10 } } \right\rangle$$
D)​ $\left\langle \frac { 17 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$
E)​ $\left\langle \frac { 17 } { \sqrt { 10 } } , \frac { 17 } { \sqrt { 10 } } \right\rangle$