Question 1

Find 2u - 3v.? $\mathbf { u } = 6 \mathbf { i } , \mathbf { v } = \mathbf { j }$ ?

Accepted Answer

A)  $12 \mathbf { i } - 3 \mathbf { j }$ 
B) ? $- 12 \mathbf { i } - 3 \mathbf { j }$ 
C) ? $6 \mathbf { i } - 3 \mathbf { j }$ 
D) ? $12 \mathbf { i } + 3 \mathbf { j }$ 
E) ? $6 i - j$ 
A)  $12 \mathbf { i } - 3 \mathbf { j }$ 
B) ? $- 12 \mathbf { i } - 3 \mathbf { j }$ 
C) ? $6 \mathbf { i } - 3 \mathbf { j }$ 
D) ? $12 \mathbf { i } + 3 \mathbf { j }$ 
E) ? $6 i - j$

Question 2

Find a unit vector in the direction of the given vector.? $\mathbf { u } = \langle - 6,6 \rangle$ ?

Accepted Answer

A)  $\langle 0,0 \rangle$ 
B) ? $\left\langle - \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right\rangle$ 
C) ? $\left\langle - \frac { \sqrt { 2 } } { 2 } , - \frac { \sqrt { 2 } } { 2 } \right\rangle$ 
D) ? $\left\langle \frac { \sqrt { 2 } } { 2 } , - \frac { \sqrt { 2 } } { 2 } \right\rangle$ 
E) ? $\left\langle \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right\rangle$ 
A)  $\langle 0,0 \rangle$ 
B) ? $\left\langle - \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right\rangle$ 
C) ? $\left\langle - \frac { \sqrt { 2 } } { 2 } , - \frac { \sqrt { 2 } } { 2 } \right\rangle$ 
D) ? $\left\langle \frac { \sqrt { 2 } } { 2 } , - \frac { \sqrt { 2 } } { 2 } \right\rangle$ 
E) ? $\left\langle \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right\rangle$

Question 3

Find the component form and the magnitude of the vector v. Initial Point
Terminal Point
(1,3)
(-8,-9)

Accepted Answer

A)  $\mathbf { v } = \langle 1 , - 8 \rangle ; \| \mathbf { v } \| = 15$ 
B)  $\mathbf { v } = \langle - 9 , - 12 \rangle ; \| \mathbf { v } \| = 8$ 
C) ? $\mathbf { v } = \langle - 9 , - 12 \rangle ; \| \mathbf { v } \| = 15$ 
D) ? $\mathbf { v } = \langle - 8 , - 9 \rangle ; \| \mathbf { v } \| = 12$ 
E) ? $\mathbf { v } = \langle 1 , - 8 \rangle ; \| \mathbf { v } \| = 8$ 
A)  $\mathbf { v } = \langle 1 , - 8 \rangle ; \| \mathbf { v } \| = 15$ 
B)  $\mathbf { v } = \langle - 9 , - 12 \rangle ; \| \mathbf { v } \| = 8$ 
C) ? $\mathbf { v } = \langle - 9 , - 12 \rangle ; \| \mathbf { v } \| = 15$ 
D) ? $\mathbf { v } = \langle - 8 , - 9 \rangle ; \| \mathbf { v } \| = 12$ 
E) ? $\mathbf { v } = \langle 1 , - 8 \rangle ; \| \mathbf { v } \| = 8$

Question 4

Select a linear combination of the standard unit vectors i and j of given initial and terminal points of a vector. Initial Point
Terminal Point
(-9,7)
​(0,2)
​

Accepted Answer

A) -9i - 5j 
B) ​-5i + 9j 
C) ​​-5i ​- 9j 
D) ​9i - 5j 
E) ​9i + 5j 
A) -9i - 5j 
B) ​-5i + 9j 
C) ​​-5i ​- 9j 
D) ​9i - 5j 
E) ​9i + 5j

Question 5

Find the component form of v if $\| \mathbf { v } \| = 8$ and the angle it makes with the x-axis is 30°.

Accepted Answer

A) ? $\langle 4 \sqrt { 3 } , 4 \rangle$ 
B) ? $\langle 4,4 \sqrt { 3 } \rangle$ 
C) ? $\langle 8,8 \sqrt { 3 } \rangle$ 
D) ? $\langle 8 \sqrt { 3 } , 8 \rangle$ 
E) ? $\langle 4 \sqrt { 2 } , 4 \sqrt { 2 } \rangle$ 
A) ? $\langle 4 \sqrt { 3 } , 4 \rangle$ 
B) ? $\langle 4,4 \sqrt { 3 } \rangle$ 
C) ? $\langle 8,8 \sqrt { 3 } \rangle$ 
D) ? $\langle 8 \sqrt { 3 } , 8 \rangle$ 
E) ? $\langle 4 \sqrt { 2 } , 4 \sqrt { 2 } \rangle$

Question 6

Find u + v.? $\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 0,0 \rangle$ ?

Accepted Answer

A) ? $$\langle - 4,4 \rangle$$ 
B)  $\langle - 5 , - 5 \rangle$ ? 
C) ? $\langle 3,3 \rangle$ 
D) ? $\langle - 5,3 \rangle$ 
E) ? $\langle 3 , - 5 \rangle$ 
A) ? $$\langle - 4,4 \rangle$$ 
B)  $\langle - 5 , - 5 \rangle$ ? 
C) ? $\langle 3,3 \rangle$ 
D) ? $\langle - 5,3 \rangle$ 
E) ? $\langle 3 , - 5 \rangle$

Question 7

A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at θ degrees from the horizontal. ​
Find W if F = 620 pounds and θ = 20°.Approximate the answer to one decimal place.
​

Accepted Answer

A) 1,812.8 lb 
B) 1,810.8 lb 
C) 1,814.8 lb 
D) 1,811.8 lb 
E) 1,813.8 lb 
A) 1,812.8 lb 
B) 1,810.8 lb 
C) 1,814.8 lb 
D) 1,811.8 lb 
E) 1,813.8 lb

Question 8

?Find a unit vector in the direction of $\mathbf { u } = 4 \mathbf { i } - 2 \mathbf { j }$ . ?

Accepted Answer

A) ? $\left\langle - \frac { 2 \sqrt { 5 } } { 5 } , \frac { \sqrt { 5 } } { 5 } \right\rangle$ 
B) ? $\left\langle \frac { 4 \sqrt { 5 } } { 5 } , - \frac { 2 \sqrt { 5 } } { 5 } \right\rangle$ 
C) ? $\left\langle \frac { 8 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right\rangle$ 
D) ? $\left\langle \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 2 \sqrt { 5 } } { 5 } \right\rangle$ 
E) ? $\left\langle \frac { 2 \sqrt { 5 } } { 5 } , - \frac { \sqrt { 5 } } { 5 } \right\rangle$ 
A) ? $\left\langle - \frac { 2 \sqrt { 5 } } { 5 } , \frac { \sqrt { 5 } } { 5 } \right\rangle$ 
B) ? $\left\langle \frac { 4 \sqrt { 5 } } { 5 } , - \frac { 2 \sqrt { 5 } } { 5 } \right\rangle$ 
C) ? $\left\langle \frac { 8 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right\rangle$ 
D) ? $\left\langle \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 2 \sqrt { 5 } } { 5 } \right\rangle$ 
E) ? $\left\langle \frac { 2 \sqrt { 5 } } { 5 } , - \frac { \sqrt { 5 } } { 5 } \right\rangle$

Question 9

Select a linear combination of the standard unit vectors i and j of given initial and terminal points of a vector.. ​
Initial Point
Terminal Point
(0,-8)
​(9,12)
​
​

Accepted Answer

A) ​9i + 20j 
B) ​-9i - 20j 
C) ​20i - 9j 
D) ​​20i + 9j 
E) ​9i - 20j 
A) ​9i + 20j 
B) ​-9i - 20j 
C) ​20i - 9j 
D) ​​20i + 9j 
E) ​9i - 20j

Question 10

Find u - v.? $\mathbf { u } = \mathbf { i } + \mathbf { j } , \mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j }$ ?

Accepted Answer

A) ?-i - 4j 
B) -2i - 4j 
C) -i + 4j 
D) -2i + 4j 
E) -i + 5j 
A) ?-i - 4j 
B) -2i - 4j 
C) -i + 4j 
D) -2i + 4j 
E) -i + 5j

Question 11

?Find a unit vector in the direction of $\mathbf { w } = \langle - 4 , - 6 \rangle$ . ?

Accepted Answer

A) ? $\left\langle - \frac { 2 \sqrt { 13 } } { 13 } , - \frac { 3 \sqrt { 13 } } { 13 } \right\rangle$ 
B) ? $\left\langle - \frac { \sqrt { 13 } } { 39 } , - \frac { \sqrt { 13 } } { 26 } \right\rangle$ 
C) ? $\left\langle - \frac { \sqrt { 13 } } { 4 } , - \frac { \sqrt { 13 } } { 6 } \right\rangle$ 
D) ? $\left\langle - \frac { \sqrt { 13 } } { 13 } , - \frac { \sqrt { 13 } } { 13 } \right\rangle$ 
E) ? $\left\langle - \frac { 8 \sqrt { 13 } } { 13 } , - \frac { 12 \sqrt { 13 } } { 13 } \right\rangle$ 
A) ? $\left\langle - \frac { 2 \sqrt { 13 } } { 13 } , - \frac { 3 \sqrt { 13 } } { 13 } \right\rangle$ 
B) ? $\left\langle - \frac { \sqrt { 13 } } { 39 } , - \frac { \sqrt { 13 } } { 26 } \right\rangle$ 
C) ? $\left\langle - \frac { \sqrt { 13 } } { 4 } , - \frac { \sqrt { 13 } } { 6 } \right\rangle$ 
D) ? $\left\langle - \frac { \sqrt { 13 } } { 13 } , - \frac { \sqrt { 13 } } { 13 } \right\rangle$ 
E) ? $\left\langle - \frac { 8 \sqrt { 13 } } { 13 } , - \frac { 12 \sqrt { 13 } } { 13 } \right\rangle$

Question 12

A loaded barge is being towed by two tugboats,and the magnitude of the resultant is 6,400 pounds directed along the axis of the barge (see figure).Find the tension in the tow lines if they each make an angle 21° with the axis of the barge.Approximate the answer to one decimal place.​   ​ where a = 21°.

Accepted Answer

A) 3,425.7 lb 
B) 3,429.7 lb 
C) 3,426.7 lb 
D) 3,428.7 lb 
E) 3,427.7 lb 
A) 3,425.7 lb 
B) 3,429.7 lb 
C) 3,426.7 lb 
D) 3,428.7 lb 
E) 3,427.7 lb

Question 13

Three forces with magnitudes of 74 pounds,98 pounds,and 105 pounds act on an object at angles 170°,220°,and 330°,respectively,with the positive x-axis.Find the magnitude and direction of the resultant force.Round answers to two decimal places.

Accepted Answer

A) 99.08 pounds;240.95° 
B) 99.08 pounds;60.95° 
C) 117.42 pounds;240.95° 
D) 139.13 pounds;272.47° 
E) 117.42 pounds;60.95° 
A) 99.08 pounds;240.95° 
B) 99.08 pounds;60.95° 
C) 117.42 pounds;240.95° 
D) 139.13 pounds;272.47° 
E) 117.42 pounds;60.95°

Question 14

Given $\mathbf { u } = 5 \mathbf { i } - 3 \mathbf { j }$ and $\mathbf { v } = \mathbf { i } + 2 \mathbf { j }$ ,determine $9 \mathbf { u } - 3 \mathbf { v }$ .

Accepted Answer

A) ? $9 \mathbf { u } - 3 \mathbf { v } = 39 \mathrm { i } - 39 \mathrm { j }$ 
B) ? $9 \mathbf { u } - 3 \mathbf { v } = 3 \mathbf { i } + 3 \mathbf { j }$ 
C) ? $9 \mathbf { u } - 3 \mathbf { v } = 42 \mathbf { i } - 33 \mathbf { j }$ 
D) ? $9 \mathbf { u } - 3 \mathbf { v } = - 6 \mathbf { i } + 39 \mathbf { j }$ 
E) ? $9 \mathbf { u } - 3 \mathbf { v } = - 30 \mathbf { i } + 39 \mathbf { j }$ 
A) ? $9 \mathbf { u } - 3 \mathbf { v } = 39 \mathrm { i } - 39 \mathrm { j }$ 
B) ? $9 \mathbf { u } - 3 \mathbf { v } = 3 \mathbf { i } + 3 \mathbf { j }$ 
C) ? $9 \mathbf { u } - 3 \mathbf { v } = 42 \mathbf { i } - 33 \mathbf { j }$ 
D) ? $9 \mathbf { u } - 3 \mathbf { v } = - 6 \mathbf { i } + 39 \mathbf { j }$ 
E) ? $9 \mathbf { u } - 3 \mathbf { v } = - 30 \mathbf { i } + 39 \mathbf { j }$

Question 15

Find u - v.? $\mathbf { u } = \langle 0,0 \rangle , \mathbf { v } = \langle 2,1 \rangle$ ?

Accepted Answer

A) ? $\langle - 1 , - 2 \rangle$ 
B) ? $\langle - 1 , - 1 \rangle$ 
C)  $\langle - 2 , - 1 \rangle$ 
D) ? $\langle - 2 , - 2 \rangle$ 
E) ? $\langle - 1,0 \rangle$ 
A) ? $\langle - 1 , - 2 \rangle$ 
B) ? $\langle - 1 , - 1 \rangle$ 
C)  $\langle - 2 , - 1 \rangle$ 
D) ? $\langle - 2 , - 2 \rangle$ 
E) ? $\langle - 1,0 \rangle$

Question 16

?Find the magnitude and direction angle of $\mathbf { v } = - 2 \mathbf { i } - 6 \mathbf { j }$ .Round the direction angle to the nearest degree.

Accepted Answer

A) ??||v|| = $3 \sqrt { 7 }$ ;? = 269° 
B) ??||v|| = $\sqrt { 30 }$ ;? = 239° 
C) ??||v|| = $2 \sqrt { 13 }$ ;? = 260° 
D) ?||v|| = $2 \sqrt { 10 }$ ;? = 252° 
E) ??||v|| = $\sqrt { 67 }$ ;? = 279° 
A) ??||v|| = $3 \sqrt { 7 }$ ;? = 269° 
B) ??||v|| = $\sqrt { 30 }$ ;? = 239° 
C) ??||v|| = $2 \sqrt { 13 }$ ;? = 260° 
D) ?||v|| = $2 \sqrt { 10 }$ ;? = 252° 
E) ??||v|| = $\sqrt { 67 }$ ;? = 279°

Question 17

Find the component form of v if $\| \mathbf { v } \| = 10$ and the angle it makes with the x-axis is 120°.

Accepted Answer

A) ? $\langle - 5 \sqrt { 3 } , 5 \rangle$ 
B) ? $\langle - 5 \sqrt { 3 } , - 10 \rangle$ 
C) ? $\langle - 5 \sqrt { 2 } , 5 \sqrt { 2 } \rangle$ 
D) ? $\langle - 5,5 \sqrt { 3 } \rangle$ 
E) ? $\langle - 10,10 \sqrt { 3 } \rangle$ 
A) ? $\langle - 5 \sqrt { 3 } , 5 \rangle$ 
B) ? $\langle - 5 \sqrt { 3 } , - 10 \rangle$ 
C) ? $\langle - 5 \sqrt { 2 } , 5 \sqrt { 2 } \rangle$ 
D) ? $\langle - 5,5 \sqrt { 3 } \rangle$ 
E) ? $\langle - 10,10 \sqrt { 3 } \rangle$

Question 18

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. ?
Magnitude
Angle $\| \mathbf { v } \| = 4$ 0°
?
?

Accepted Answer

A)  $\langle 0,0 \rangle$ 
B) ? $\langle 4,4 \rangle$ 
C) ? $\langle 0,4 \rangle$ 
D) ? $\langle 4,0 \rangle$ 
E) ? $\langle - 4,0 \rangle$ 
A)  $\langle 0,0 \rangle$ 
B) ? $\langle 4,4 \rangle$ 
C) ? $\langle 0,4 \rangle$ 
D) ? $\langle 4,0 \rangle$ 
E) ? $\langle - 4,0 \rangle$

Question 19

?If $\mathbf { u } = - 3 \mathbf { i } + 6 \mathbf { j }$ and $\mathbf { v } = - 6 \mathbf { i } + 5 \mathbf { j }$ ,find $\mathbf { w } = 3 \mathbf { u } + 5 \mathbf { v }$ .

Accepted Answer

A) ? $\mathbf { w } = - 39 \mathbf { i } + 43 \mathbf { j }$ 
B) ? $\mathbf { w } = - 33 \mathbf { i } + 43 \mathbf { j }$ 
C) ? $\mathbf { w } = - 27 \mathbf { i } + 55 \mathbf { j }$ 
D) ? $\mathbf { w } = - 39 \mathbf { i } + 45 \mathbf { j }$ 
E) ? $\mathbf { w } = 16 \mathbf { i } - 12 \mathbf { j }$ 
A) ? $\mathbf { w } = - 39 \mathbf { i } + 43 \mathbf { j }$ 
B) ? $\mathbf { w } = - 33 \mathbf { i } + 43 \mathbf { j }$ 
C) ? $\mathbf { w } = - 27 \mathbf { i } + 55 \mathbf { j }$ 
D) ? $\mathbf { w } = - 39 \mathbf { i } + 45 \mathbf { j }$ 
E) ? $\mathbf { w } = 16 \mathbf { i } - 12 \mathbf { j }$

Question 20

Find the component form of vector v. ??   ?

Accepted Answer

A)  $\mathbf { v } = \langle 4 , - 5 \rangle$ 
B) ? $$\mathbf { v } = \langle - 4 , - 5 \rangle$$ 
C) ? $\mathbf { v } = \langle 4 , - 3 \rangle$ 
D) ? $$\mathbf { v } = \langle 5,4 \rangle$$ 
E) ? $\mathbf { v } = \langle - 5,4 \rangle$ 
A)  $\mathbf { v } = \langle 4 , - 5 \rangle$ 
B) ? $$\mathbf { v } = \langle - 4 , - 5 \rangle$$ 
C) ? $\mathbf { v } = \langle 4 , - 3 \rangle$ 
D) ? $$\mathbf { v } = \langle 5,4 \rangle$$ 
E) ? $\mathbf { v } = \langle - 5,4 \rangle$

Exam 38:Vectors in the Plane

Find 2u - 3v.? u=6i,v=j\mathbf { u } = 6 \mathbf { i } , \mathbf { v } = \mathbf { j }u=6i,v=j ?

Find a unit vector in the direction of the given vector.? u=⟨−6,6⟩\mathbf { u } = \langle - 6,6 \rangleu=⟨−6,6⟩ ?

Find the component form and the magnitude of the vector v. Initial Point Terminal Point (1,3) (-8,-9)

Select a linear combination of the standard unit vectors i and j of given initial and terminal points of a vector. Initial Point Terminal Point (-9,7) ​(0,2) ​

Find the component form of v if ∥v∥=8\| \mathbf { v } \| = 8∥v∥=8 and the angle it makes with the x-axis is 30°.

Find u + v.? u=⟨−5,3⟩,v=⟨0,0⟩\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 0,0 \rangleu=⟨−5,3⟩,v=⟨0,0⟩ ?

A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at θ degrees from the horizontal. ​ Find W if F = 620 pounds and θ = 20°.Approximate the answer to one decimal place. ​

?Find a unit vector in the direction of u=4i−2j\mathbf { u } = 4 \mathbf { i } - 2 \mathbf { j }u=4i−2j . ?

Select a linear combination of the standard unit vectors i and j of given initial and terminal points of a vector.. ​ Initial Point Terminal Point (0,-8) ​(9,12) ​ ​

Find u - v.? u=i+j,v=2i−3j\mathbf { u } = \mathbf { i } + \mathbf { j } , \mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j }u=i+j,v=2i−3j ?

?Find a unit vector in the direction of w=⟨−4,−6⟩\mathbf { w } = \langle - 4 , - 6 \ranglew=⟨−4,−6⟩ . ?

Three forces with magnitudes of 74 pounds,98 pounds,and 105 pounds act on an object at angles 170°,220°,and 330°,respectively,with the positive x-axis.Find the magnitude and direction of the resultant force.Round answers to two decimal places.

Given u=5i−3j\mathbf { u } = 5 \mathbf { i } - 3 \mathbf { j }u=5i−3j and v=i+2j\mathbf { v } = \mathbf { i } + 2 \mathbf { j }v=i+2j ,determine 9u−3v9 \mathbf { u } - 3 \mathbf { v }9u−3v .

Find u - v.? u=⟨0,0⟩,v=⟨2,1⟩\mathbf { u } = \langle 0,0 \rangle , \mathbf { v } = \langle 2,1 \rangleu=⟨0,0⟩,v=⟨2,1⟩ ?

?Find the magnitude and direction angle of v=−2i−6j\mathbf { v } = - 2 \mathbf { i } - 6 \mathbf { j }v=−2i−6j .Round the direction angle to the nearest degree.

Find the component form of v if ∥v∥=10\| \mathbf { v } \| = 10∥v∥=10 and the angle it makes with the x-axis is 120°.

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. ? Magnitude Angle ∥v∥=4\| \mathbf { v } \| = 4∥v∥=4 0° ? ?

?If u=−3i+6j\mathbf { u } = - 3 \mathbf { i } + 6 \mathbf { j }u=−3i+6j and v=−6i+5j\mathbf { v } = - 6 \mathbf { i } + 5 \mathbf { j }v=−6i+5j ,find w=3u+5v\mathbf { w } = 3 \mathbf { u } + 5 \mathbf { v }w=3u+5v .

Find the component form of vector v. ?? ?

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses