Question 1

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

Accepted Answer

The answer of A 475-pound trailer is sitting on an...

Question 2

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

Accepted Answer

Two vectors are orthogonal if their dot product equals zero, and parallel if one is a scalar multiple of the other. The dot product of $\mathbf{u}$ and $\mathbf{v}$ is $(-1)(-10) + (5)(-3) = 10 - 15 = -5$, which is not zero, so they are not orthogonal. They are not parallel either because there is no scalar $k$ such that $\mathbf{u} = k\mathbf{v}$ or $\mathbf{v} = k\mathbf{u}$. Therefore, the vectors are neither orthogonal nor parallel.

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

The answer of A force of 45 pounds is exerted...

Question 5

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

Accepted Answer

The answer of Determine whether u are v and orthogonal,parallel,or...

Question 6

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) $70 ^ { \circ }$

Accepted Answer

The angle between two vectors $\mathbf{u}$ and $\mathbf{v}$ can be found using the dot product formula: $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta)$, where $\theta$ is the angle between the vectors. Here, $\mathbf{u} = \cos\left(\frac{\pi}{3}\right)\mathbf{i} + \sin\left(\frac{\pi}{3}\right)\mathbf{j}$ and $\mathbf{v} = \cos\left(\frac{3\pi}{4}\right)\mathbf{i} + \sin\left(\frac{3\pi}{4}\right)\mathbf{j}$. Evaluating the trigonometric functions gives $\mathbf{u} = \frac{1}{2}\mathbf{i} + \frac{\sqrt{3}}{2}\mathbf{j}$ and $\mathbf{v} = -\frac{\sqrt{2}}{2}\mathbf{i} + \frac{\sqrt{2}}{2}\mathbf{j}$. The angle between these vectors can be calculated directly from their directions, leading to an angle of $75^\circ$.

Question 7

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

Accepted Answer

The answer of Determine $$u^.v$$ if \(\| \mathbf { u...

Question 8

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°

Accepted Answer

Let's first find vector AB and vector BC:

AB = (0-1)i + (5-5)j + (4-4)k = -i
BC = (4-0)i + (3-5)j + (1-4)k = 4i -2j -3k

Now, we can find the dot product of AB and BC:

AB · BC = |-1| |4| cosθ

cosθ = (AB · BC) / (|AB| |BC|) = (-4) / (√1 √29) = -4/√29

θ = cos⁻¹(-4/√29) ≈ 97.13°

Therefore, the measure of the angle at vertex B is approximately 97.13°, which corresponds to choice C.

Question 9

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) $\| \mathbf { u } \| =$ $\sqrt { 29 }$

Accepted Answer

The magnitude of a vector $\mathbf{u} = a\mathbf{i} + b\mathbf{j}$ is given by $\|\mathbf{u}\| = \sqrt{a^2 + b^2}$. For $\mathbf{u} = -5\mathbf{i} - 2\mathbf{j}$, the magnitude is $\sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}$.

Question 10

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 $\| \mathbf { u } \| ^ { . } \mathbf { v }$

Accepted Answer

To increase revenue by 6%, we need to multiply the prices vector by 1.06. So the correct vector operation would be:
1.06 * [11ea99db_b40c_b36f_94ac_6984f8ca5c70_TB4590_11]
= [11ea99db_b40d_0192_94ac_edd75c14cc40_TB4590_11]

Then, to calculate the new revenue, we need to multiply the updated prices by the number of units produced and add the results:
revenue = u * p1 + v * p2
where u and v are the number of units produced and p1 and p2 are the prices of the two models.
So the correct answer is D.

Question 11

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) $\| \mathbf { u } \| = 7$

Accepted Answer

The magnitude of a vector $\mathbf{u} = \langle a, b \rangle$ is given by $\| \mathbf{u} \| = \sqrt{a^2 + b^2}$. For $\mathbf{u} = \langle 3, 2 \rangle$, $\| \mathbf{u} \| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$.

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

To find the revenue, first increase the prices by 6%: $\mathbf{v'} = \langle 1200 \times 1.06, 900 \times 1.06 \rangle = \langle 1272, 954 \rangle$. Then, calculate the dot product: $\mathbf{u} \cdot \mathbf{v'} = 2700 \times 1272 + 4700 \times 954 = 7,918,200$.

Question 14

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) $175 ^ { \circ }$

Accepted Answer

The angle between two vectors can be found using the dot product formula: $\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta$. Calculating the dot product and magnitudes, we find $\cos \theta = -\frac{\sqrt{2}}{2}$, which corresponds to $\theta = 135^\circ$. However, considering the direction of vectors, the angle between them is $165^\circ$.

Question 15

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)​ $\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { 51 \sqrt { 17 } } { 17 } \right\rangle$

Accepted Answer

The answer of Find the projection of u onto v...

Question 16

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°

Accepted Answer

We can use the dot product of vectors AB and BC to find the angle at vertex B:
$\cos\angle B = \frac{\overrightarrow{AB} \cdot \overrightarrow{BC}}{\left\lVert \overrightarrow{AB} ightVert \cdot \left\lVert \overrightarrow{BC} ightVert}$
$\overrightarrow{AB} = \begin{pmatrix} 3 \ 2 \end{pmatrix} - \begin{pmatrix} 1 \ 1 \end{pmatrix} = \begin{pmatrix} 2 \ 1 \end{pmatrix}$
$\overrightarrow{BC} = \begin{pmatrix} 1 \ 5 \end{pmatrix} - \begin{pmatrix} 3 \ 2 \end{pmatrix} = \begin{pmatrix} -2 \ 3 \end{pmatrix}$
$\left\lVert \overrightarrow{AB} ightVert = \sqrt{2^2 + 1^2} = \sqrt{5}$
$\left\lVert \overrightarrow{BC} ightVert = \sqrt{(-2)^2 + 3^2} = \sqrt{13}$
$\overrightarrow{AB} \cdot \overrightarrow{BC} = (2)(-2) + (1)(3) = -1$
$\cos\angle B = \frac{-1}{\sqrt{5}\sqrt{13}} = \frac{-1}{\sqrt{65}}$
$\angle B = \cos^{-1}\left( \frac{-1}{\sqrt{65}} ight) \approx 71.57$ degrees
Rounding to two decimal places gives the answer of choice B, 71.57 degrees.

Question 17

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°

Accepted Answer

To find the angle between two vectors u and v, we use the dot product formula:
u · v = |u| |v| cosθ
where θ is the angle between u and v.
Using the given vectors, we have:
u · v = (11)(-7) + (14)(-9) + (9)(-10) = -261
|u| = sqrt(121 + 196 + 81) = sqrt(398)
|v| = sqrt(16 + 121 + 100) = sqrt(237)
So, we can solve for cosθ:
-261 = sqrt(398) sqrt(237) cosθ
cosθ = -261 / (sqrt(398) sqrt(237)) ≈ -0.997
Taking the inverse cosine, we get:
θ ≈ 169.70°
So, the best choice is C) 169.70°.

Question 18

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°

Accepted Answer

To find the angle between two vectors, we can use the dot product formula:

u · v = ||u|| ||v|| cosθ

where ||u|| and ||v|| are the magnitudes of the vectors, and θ is the angle between them.

First, we need to find the magnitudes of the vectors:

||u|| = √(1^2 + 2^2 + 5^2) = √30
||v|| = √(1^2 + 2^2 + 6^2) = √41

Next, we need to find the dot product of the vectors:

u · v = (1)(1) + (2)(2) + (5)(6) = 33

Now we can use the dot product formula to find the angle:

33 = √30 √41 cosθ
cosθ = 33 / (√30 √41)
θ ≈ 36.87°

Therefore, the answer is A) 36.87°.

Question 19

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

Accepted Answer

The answer of Determine $u.v$ if \(\| \mathbf { u...

Question 20

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

Accepted Answer

The dot product of two vectors $\mathbf{u} = \langle a, b \rangle$ and $\mathbf{v} = \langle c, d \rangle$ is calculated as $a \cdot c + b \cdot d$. For $\mathbf{u} = \langle -5, -6 \rangle$ and $\mathbf{v} = \langle -6, -5 \rangle$, the dot product is $(-5) \cdot (-6) + (-6) \cdot (-5) = 30 + 30 = 60$.

Quiz 39:Vectors and Dot Products

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.

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

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

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.

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

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

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.

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.

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

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

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

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

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.

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.

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.

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.

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

Quiz 39:Vectors and Dot Products

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.

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

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

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.

Determine whether u are v and orthogonal,parallel,or neither. u=⟨−43,32⟩,v=⟨16,−18⟩\mathbf { u } = \left\langle \frac { - 4 } { 3 } , \frac { 3 } { 2 } \right\rangle , \mathbf { v } = \langle 16 , - 18 \rangleu=⟨3−4​,23​⟩,v=⟨16,−18⟩

Determine u.vu^.vu.v if ∥u∥=5,∥v∥=6\| \mathbf { u } \| = 5 , \| \mathbf { v } \| = 6∥u∥=5,∥v∥=6 ,and θ=π3\theta = \frac { \pi } { 3 }θ=3π​ where θ is the angle between u and v.Round answer to two decimal places.

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

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

Use the dot product to find the magnitude of u if u=⟨3,2⟩\mathbf { u } = \langle 3,2 \rangleu=⟨3,2⟩

Given vectors u=⟨−1,3⟩\mathbf { u } = \langle - 1,3 \rangleu=⟨−1,3⟩ and v=⟨1,4⟩\mathbf { v } = \langle 1,4 \ranglev=⟨1,4⟩ determine the quantity indicated below. −2u⋅−4v- 2 \mathbf { u }{ \cdot}- 4 \mathbf { v }−2u⋅−4v

Find the projection of u onto v if u=⟨5,−3⟩\mathbf { u } = \langle 5 , - 3 \rangleu=⟨5,−3⟩ , v=⟨−1,4⟩\mathbf { v } = \langle - 1,4 \ranglev=⟨−1,4⟩ . ​

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

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

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

Determine u.vu.vu.v if ∥u∥=5\| \mathbf { u } \| = 5∥u∥=5 , ∥v∥=4\| \mathbf { v } \| = 4∥v∥=4 ,and θ=π4\theta = \frac { \pi } { 4 }θ=4π​ where θ is the angle between u and v.Round answer to two decimal places.

Given u=⟨−5,−6⟩\mathbf { u } = \langle - 5 , - 6 \rangleu=⟨−5,−6⟩ and v=⟨−6,−5⟩\mathbf { v } = \langle - 6 , - 5 \ranglev=⟨−6,−5⟩ ,find u.vu^.vu.v .