Question 1

Let w be a vector with initial point (4,-4)and terminal point (2,5).Write w as a linear combination of the standard unit vectors i and j.
A)? $\mathbf { w } = 8 \mathbf { i } - 3 \mathbf { j }$
B)? $\mathbf { w } = 0 \mathbf { i } + 3 \mathbf { j }$
C)? $\mathbf { w } = \mathbf { i } + 6 \mathbf { j }$
D)? $\mathbf { w } = - 2 \mathbf { i } + 9 \mathbf { j }$
E)? $\mathbf { w } = 6 \mathbf { i } - 9 \mathbf { j }$

Accepted Answer

To find the vector $ \mathbf{w} $ as a linear combination of the standard unit vectors $ \mathbf{i} $ and $ \mathbf{j} $, we subtract the coordinates of the initial point from the coordinates of the terminal point: $ \mathbf{w} = (2 - 4) \mathbf{i} + (5 - (-4)) \mathbf{j} = -2 \mathbf{i} + 9 \mathbf{j} $.

Question 2

?Find a unit vector in the direction of $\mathbf { u } = 4 \mathbf { i } - 2 \mathbf { j }$ . ?
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)? $\left\langle \frac { 2 \sqrt { 5 } } { 5 } , - \frac { \sqrt { 5 } } { 5 } \right\rangle$

Accepted Answer

The answer of ?Find a unit vector in the direction...

Question 3

?Find the magnitude of vector v with initial point (7,5)and terminal point (-3,-5). ?
A)? $10 \sqrt { 2 }$
B)?14
C)? $20 \sqrt { 2 }$
D)? $\sqrt { 74 }$
E)? $2 \sqrt { 51 }$

Accepted Answer

The magnitude of vector v is calculated using the distance formula: $\sqrt{(-3-7)^2 + (-5-5)^2} = \sqrt{(-10)^2 + (-10)^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2}$.

Question 4

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 F if W = 5,700 pounds and θ = 29°.Approximate the answer to one decimal place.
​
A)2,761.4 lb
B)2,764.4 lb
C)2,763.4 lb
D)2,765.4 lb
E)2,762.4 lb

Accepted Answer

The force required to pull an object up an inclined plane is given by the formula $F = W \sin(\theta)$, where $F$ is the force in pounds, $W$ is the weight of the object in pounds, and $\theta$ is the angle of inclination in degrees. For $W = 5,700$ pounds and $\theta = 29^\circ$, we calculate $F = 5,700 \times \sin(29^\circ)$. Using a calculator, $\sin(29^\circ) \approx 0.4848$, so $F \approx 5,700 \times 0.4848 \approx 2,763.36$ pounds, which rounds to $2,763.4$ pounds.

Question 5

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°.
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

Accepted Answer

The answer of A loaded barge is being towed by...

Question 6

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 F if W = 100 pounds and θ = 11°.Approximate the answer to one decimal place.
​
A)18.1 lb
B)21.1 lb
C)19.1 lb
D)20.1 lb
E)17.1 lb

Accepted Answer

The force required to pull the object up the ramp is given by F = W * sin(θ). For W = 100 pounds and θ = 11°, F = 100 * sin(11°) ≈ 19.1 lb.

Question 7

A traffic light weighing 15 pounds is suspended by two cables (see figure).Find the tension in each cable if θ₁ = 40° and θ₂ = 35°.Approximate the answers to one decimal place. A)T_L = 12.7 lb;TR = 12.7 lb B)T_L = 11.9 lb;TR = 11.9 lb C)T_L = 11.9 lb;TR = 12.7 lb D)T_L = 12.7 lb;TR = 11.9 lb E)T_L = 13.7 lb;TR = 12.9 lb

Accepted Answer

Using the law of sines, we can write:

TS1U1B1/ sin(40°) = TR/ sin(145°)
TS1U1B1/ sin(35°) = TR/ sin(125°)

Solving for TS1U1B1 and TR, we get:

TS1U1B1 = sin(40°) sin(125°) TR / sin(145°) sin(35°)

TR = sin(35°) sin(145°) TS1U1B1 / sin(125°) sin(40°)

Plugging in the given values and simplifying, we get:

TS1U1B1 ≈ 12.7 lb

TR ≈ 11.9 lb

Therefore, the best choice is (D), which gives the correct values for both tension forces.

Question 8

Using the figure below,sketch a graph of the given vector.[The graphs in the answer choices are drawn to the same scale as the graph below.]? $-u$  ?
A)? 
B)? 
C)? 
D)? 
E)none of these

Accepted Answer

The answer of Using the figure below,sketch a graph of...

Question 9

The initial and terminal points of vector v are (3,-8)and (7,1),respectively.Select v in component form. ?
A)? $\mathbf { v } = \langle - 4 , - 9 \rangle$
B)? $\mathbf { v } = \langle - 4,9 \rangle$
C) $\mathbf { v } = \langle 4,9 \rangle$
D) $\mathbf { v } = \langle 4 , - 9 \rangle$
E)? $\mathbf { v } = \langle 9,4 \rangle$

Accepted Answer

The component form of a vector is found by subtracting the coordinates of the initial point from the coordinates of the terminal point. Thus, $v = \langle 7 - 3, 1 - (-8) \rangle = \langle 4, 9 \rangle$.

Question 10

Find the magnitude of vector v.? 
A) $\| \mathbf { v } \| = 5$
B)? $\| \mathbf { v } \| = 2 \sqrt { 7 }$
C)? $\| \mathbf { v } \| = 6$
D)? $\| \mathbf { v } \| = 4 \sqrt { 2 }$
E)? $\| \mathbf { v } \| = 6 \sqrt { 3 }$

Accepted Answer

The answer of Find the magnitude of vector v.? 
A)...

Question 11

Find the component form of vector v. ??  ?
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)? $\mathbf { v } = \langle - 5,4 \rangle$

Accepted Answer

The answer of Find the component form of vector v....

Question 12

Given $$\mathbf { u } = \langle 3 , - 2 \rangle$$ and $\mathbf { v } = \langle - 3,2 \rangle$ ,determine $- 4 \mathbf { u } - \mathbf { v }$ .
A)? $$- 4 \mathbf { u } - \mathbf { v } = \langle 9 , - 14 \rangle$$
B) $- 4 \mathbf { u } - \mathbf { v } = \langle - 9,6 \rangle$
C)? $- 4 \mathbf { u } - \mathbf { v } = \langle - 14 , - 14 \rangle$
D)? $- 4 \mathbf { u } - \mathbf { v } = \langle 11 , - 14 \rangle$
E)? $- 4 \mathbf { u } - \mathbf { v } = \langle - 11,10 \rangle$

Accepted Answer

The answer of Given \[\mathbf { u } = \langle...

Question 13

A traffic light weighing 16 pounds is suspended by two cables (see figure).Find the tension in each cable.Approximate the answer to one decimal place. ​​  ​
Where b = 28°
​
A)18.0 lb
B)17.0 lb
C)16.0 lb
D)19.0 lb
E)15.0 lb

Accepted Answer

We can use the equations for static equilibrium to solve for the tensions in the cables. Let T1 and T2 be the tensions in the cables, and let the weight of the traffic light be W=16 lbs. Then, we have:

ΣF_x = 0: T1*cos(b) - T2*cos(b) = 0
ΣF_y = 0: T1*sin(b) + T2*sin(b) - W = 0

Simplifying the first equation gives T1 = T2. Substituting this into the second equation and solving for T1 or T2, we get:

T1 = T2 = W/(2*sin(b)) = 16/(2*sin(28)) ≈ 17.0 lbs

Therefore the best choice is B, which is the closest answer to the calculated tension in pounds.

Question 14

?Find the vector v that has a magnitude of 9 and is in the same direction as u,where $\mathbf { u } = \langle 6 , - 5 \rangle$ . ?
A)? $\mathbf { v } = \left\langle \frac { 5 \sqrt { 61 } } { 61 } , - \frac { 6 \sqrt { 61 } } { 61 } \right\rangle$
B)? $\mathrm { v } = \left\langle \frac { 45 \sqrt { 61 } } { 61 } , - \frac { 54 \sqrt { 61 } } { 61 } \right\rangle$
C)? $\mathbf { v } = \left\langle \frac { \sqrt { 61 } } { 61 } , - \frac { \sqrt { 61 } } { 61 } \right\rangle$
D)? $\mathbf { v } = \left\langle \frac { 6 \sqrt { 61 } } { 61 } , - \frac { 5 \sqrt { 61 } } { 61 } \right\rangle$
E)? $\mathbf { v } = \left\langle \frac { 54 \sqrt { 61 } } { 61 } , - \frac { 45 \sqrt { 61 } } { 61 } \right\rangle$

Accepted Answer

The answer of ?Find the vector v that has a...

Question 15

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 θ if F = 5,100 pounds and W = 14,000 lb.Approximate the answer to one decimal place.
​
A)19.4°
B)​21.4°
C)​23.4°
D)​20.4°
E)​22.4°

Accepted Answer

The answer of A force of F pounds is required...

Question 16

Find the component form of vector v with initial point (1,-7)and terminal point (-3,1).
A)? $\mathbf { v } = \langle - 4 , - 8 \rangle$
B) $\mathbf { v } = \langle - 4,8 \rangle$
C)? $v = \langle 8 , - 4 \rangle$
D)? $v = \langle 0 , - 4 \rangle$
E)? $\mathbf { v } = \langle - 8 , - 4 \rangle$

Accepted Answer

To find the component form of vector $v$ with an initial point $P_1(x_1, y_1) = (1, -7)$ and a terminal point $P_2(x_2, y_2) = (-3, 1)$, use the formula $v = \langle x_2 - x_1, y_2 - y_1 \rangle$. Substituting the given points, $v = \langle -3 - 1, 1 - (-7) \rangle = \langle -4, 8 \rangle$.

Question 17

?Find a unit vector in the direction of $\mathbf { w } = \langle - 4 , - 6 \rangle$ . ?
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)? $\left\langle - \frac { 8 \sqrt { 13 } } { 13 } , - \frac { 12 \sqrt { 13 } } { 13 } \right\rangle$

Accepted Answer

To find a unit vector in the direction of $\mathbf{w} = \langle -4, -6 \rangle$, you first calculate the magnitude of $\mathbf{w}$, which is $\sqrt{(-4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} = \sqrt{4 \cdot 13} = 2\sqrt{13}$. Then, divide each component of $\mathbf{w}$ by its magnitude to get the unit vector: $\left\langle \frac{-4}{2\sqrt{13}}, \frac{-6}{2\sqrt{13}} \right\rangle = \left\langle -\frac{2\sqrt{13}}{13}, -\frac{3\sqrt{13}}{13} \right\rangle$.

Question 18

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.
​
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

Accepted Answer

The force required to pull an object up a ramp is given by F = W * sin(θ). Solving for W gives W = F / sin(θ). Substituting F = 620 and θ = 20°, we get W = 620 / sin(20°) ≈ 1,812.8 lb.

Question 19

Given $\mathbf { u } = 5 \mathbf { i } - 3 \mathbf { j }$ and $\mathbf { v } = \mathbf { i } + 2 \mathbf { j }$ ,determine $9 \mathbf { u } - 3 \mathbf { v }$ .
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)? $9 \mathbf { u } - 3 \mathbf { v } = - 30 \mathbf { i } + 39 \mathbf { j }$

Accepted Answer

The answer of Given \(\mathbf { u } = 5...

Question 20

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

Accepted Answer

To find the linear combination of i and j, we need to find the differences between the terminal and initial points in the x and y directions. That is:

Terminal x - Initial x = 0 - (-9) = 9
Terminal y - Initial y = 2 - 7 = -5

Thus, the linear combination is 9i - 5j, which is option D.

Quiz 38:Vectors in the Plane

Let w be a vector with initial point (4,-4) and terminal point (2,5) .Write w as a linear combination of the standard unit vectors i and j.

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

?Find the magnitude of vector v with initial point (7,5) and terminal point (-3,-5) . ?

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 F if W = 5,700 pounds and θ = 29°.Approximate the answer to one decimal place.

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 F if W = 100 pounds and θ = 11°.Approximate the answer to one decimal place.

A traffic light weighing 15 pounds is suspended by two cables (see figure) .Find the tension in each cable if θ₁ = 40° and θ₂ = 35°.Approximate the answers to one decimal place.

Using the figure below,sketch a graph of the given vector.[The graphs in the answer choices are drawn to the same scale as the graph below.]? $-u$ ?

The initial and terminal points of vector v are (3,-8) and (7,1) ,respectively.Select v in component form. ?

Find the magnitude of vector v.? $Find the magnitude of vector v.? A) \| \mathbf { v } \| = 5 B) ? \| \mathbf { v } \| = 2 \sqrt { 7 } C) ? \| \mathbf { v } \| = 6 D) ? \| \mathbf { v } \| = 4 \sqrt { 2 } E) ? \| \mathbf { v } \| = 6 \sqrt { 3 }$

Given $\mathbf { u } = \langle 3 , - 2 \rangle$ and $\mathbf { v } = \langle - 3,2 \rangle$ ,determine $- 4 \mathbf { u } - \mathbf { v }$ .

A traffic light weighing 16 pounds is suspended by two cables (see figure) .Find the tension in each cable.Approximate the answer to one decimal place. Where b = 28°

?Find the vector v that has a magnitude of 9 and is in the same direction as u,where $\mathbf { u } = \langle 6 , - 5 \rangle$ . ?

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 θ if F = 5,100 pounds and W = 14,000 lb.Approximate the answer to one decimal place.

Find the component form of vector v with initial point (1,-7) and terminal point (-3,1) .

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

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.

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

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)

Quiz 38:Vectors in the Plane

Let w be a vector with initial point (4,-4) and terminal point (2,5) .Write w as a linear combination of the standard unit vectors i and j.

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

?Find the magnitude of vector v with initial point (7,5) and terminal point (-3,-5) . ?

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 F if W = 5,700 pounds and θ = 29°.Approximate the answer to one decimal place. ​

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 F if W = 100 pounds and θ = 11°.Approximate the answer to one decimal place. ​

A traffic light weighing 15 pounds is suspended by two cables (see figure) .Find the tension in each cable if θ1 = 40° and θ2 = 35°.Approximate the answers to one decimal place.​ ​

Using the figure below,sketch a graph of the given vector.[The graphs in the answer choices are drawn to the same scale as the graph below.]? −u-u−u ?

The initial and terminal points of vector v are (3,-8) and (7,1) ,respectively.Select v in component form. ?

Find the magnitude of vector v.?

Find the component form of vector v. ?? ?

Given u=⟨3,−2⟩\mathbf { u } = \langle 3 , - 2 \rangleu=⟨3,−2⟩ and v=⟨−3,2⟩\mathbf { v } = \langle - 3,2 \ranglev=⟨−3,2⟩ ,determine −4u−v- 4 \mathbf { u } - \mathbf { v }−4u−v .

A traffic light weighing 16 pounds is suspended by two cables (see figure) .Find the tension in each cable.Approximate the answer to one decimal place. ​​ ​ Where b = 28° ​

?Find the vector v that has a magnitude of 9 and is in the same direction as u,where u=⟨6,−5⟩\mathbf { u } = \langle 6 , - 5 \rangleu=⟨6,−5⟩ . ?

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 θ if F = 5,100 pounds and W = 14,000 lb.Approximate the answer to one decimal place. ​