Question 1

Let $\mathbf { u } = \langle 6,8 \rangle$ and $\mathbf { v } = \langle 14,10 \rangle$. Find $\mathbf { u } - \mathbf { v }$
A) $\langle 16,22 \rangle$
B) $\langle - 8 , - 2 \rangle$
C) $\langle 20,18 \rangle$
D) $\langle - 4 , - 6 \rangle$
E) $\langle 22,16 \rangle$

Accepted Answer

To find $\mathbf { u } - \mathbf { v }$, subtract the corresponding components of $\mathbf { v }$ from $\mathbf { u }$: $\langle 6 - 14, 8 - 10 \rangle = \langle -8, -2 \rangle$.

Question 2

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.
$\mathbf { u } \cdot 4 \mathrm { v }$
A) vector
B) scalar

Accepted Answer

The dot product of two vectors is always a scalar, and multiplying a vector by a scalar results in a vector. Therefore, the result of $\mathbf { u } \cdot 4 \mathrm { v }$ will be a scalar multiplied by a scalar, which is still a scalar.

Question 3

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 50 feet. How much work has been Accomplished? Round answer to one decimal place.
A) 2, 250.0 foot-pounds
B) 2,598.1 foot-pounds
C) 2,099.3 foot-pounds
D) 2,348.7 foot-pounds
E) 1,948.6 foot-pounds

Accepted Answer

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

Question 4

In the figure below, $a = 9 , b = 11$, and $c = 12$. Use this information to solve the parallelogram for $\theta$. The diagonals of the parallelogram are represented by $c$ and $d$. Round answer to two decimal places.

A) $107.03 ^ { \circ }$
B) $98.52$
C) $90.00 ^ { \circ }$
D) $92.13$
E) $72.97 ^ { \circ }$

Accepted Answer

In a parallelogram, opposite angles are congruent. Let's call the angle opposite to $b$ as $\alpha$. Then, we know that $\theta + \alpha = 180 ^ {\circ}$. Using the Law of Cosines on triangle $ABE$, we have:
$$c^2 = a^2 + b^2 - 2ab\cos\theta$$
Substituting the given values, we get:
$$12^2 = 9^2 + 11^2 - 2 \cdot 9 \cdot 11 \cdot \cos\theta $$
Solving for $\cos\theta$, we get:
$$\cos\theta = \frac{9^2 + 11^2 - 12^2}{2 \cdot 9 \cdot 11} = -\frac{23}{99}$$
Since $0 ^ {\circ} \leq \theta \leq 180 ^ {\circ}$, and $\cos\theta < 0$, we have:
$$\theta \approx \cos^{-1}\left(-\frac{23}{99}\right) \approx 107.03 ^ {\circ}$$
Therefore, the answer is A.

Question 5

A plane flies 450 kilometers with a bearing of $319 ^ { \circ }$ from City A to City B (see figure below). The plane then flies 730 kilometers from City B to City C, which is directly west of City A. Find the bearing of the flight from City B to City C. Round your answer to two decimals.

A) $242.27 ^ { \circ }$
B) $332.27 ^ { \circ }$
C) $27.73 ^ { \circ }$
D) $23.85 ^ { \circ }$
E) $246.15 ^ { \circ }$

Accepted Answer

The answer of A plane flies 450 kilometers with a...

Question 6

Perform the operation below and leave the result in trigonometric form. $$\frac { 4 \left( \cos 58 ^ { \circ } + i \sin 58 ^ { \circ } \right) } { 12 \left( \cos 13 ^ { \circ } + i \sin 13 ^ { \circ } \right) }$$
A) $\frac { 1 } { 3 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$
B) $- 8 \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$
C) $16 \left( \cos 71 ^ { \circ } + i \sin 71 ^ { \circ } \right)$
D) $48 \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$
E) $\frac { 1 } { 3 } \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$

Accepted Answer

The division of complex numbers in trigonometric form is performed by dividing their magnitudes and subtracting their angles. Here, the magnitudes are $4/12 = 1/3$ and the angles subtract to give $58^\circ - 13^\circ = 45^\circ$, resulting in $\frac{1}{3}(\cos 45^\circ + i\sin 45^\circ)$.

Question 7

Let $\mathbf { w }$ be a vector with initial point $( 2,1 )$ and terminal point $( 9 , - 3 )$. Write $\mathbf { w }$ as a linear combination of the standard unit vectors $\mathbf { i }$ and $\mathbf { j }$.
A) $\mathbf { w } = \mathbf { i } + 12 \mathbf { j }$
B) $\mathbf { w } = - 5 \mathbf { i } + 8 \mathbf { j }$
C) $\mathbf { w } = 7 \mathbf { i } - 4 \mathbf { j }$
D) $\mathbf { w } = 11 \mathbf { i } + 4 \mathbf { j }$
E) $\mathbf { w } = 3 \mathbf { i } - 12 \mathbf { j }$

Accepted Answer

To find the vector $\mathbf { w }$, subtract the initial point from the terminal point: $(9 - 2)\mathbf { i } + (-3 - 1)\mathbf { j } = 7\mathbf { i } - 4\mathbf { j }$.

Question 8

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number.
$( - 6 + 6 i ) ^ { 4 }$
A) - 5,184
B) 46,656
C) 5,184
D) 186,624
E)  7,776

Accepted Answer

The answer of Use DeMoivre's Theorem to find the indicated...

Question 9

Given $a = 3 , b = 8$, and $c = 6$, use the Law of Cosines to solve the triangle for the value of $C$. Round answer to two decimal places.

A) $60.33 ^ { \circ }$
B) $39.57 ^ { \circ }$
C) $80.44 ^ { \circ }$
D) $121.86 ^ { \circ }$
E) $18.57 ^ { \circ }$

Accepted Answer

The answer of Given \(a = 3 , b =...

Question 10

Perform the operation shown below and leave the result in trigonometric form.
$$\frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }$$
A) $6 ( \cos 0.8 + i \sin 0.8 )$
B) $30 ( \cos 0.8 + i \sin 0.8 )$
C) $30 ( \cos 1.5 + i \sin 1.5 )$
D) $6 ( \cos 4.2 + i \sin 4.2 )$
E) $6 ( \cos 4.25 + i \sin 4.25 )$

Accepted Answer

The division of two complex numbers in trigonometric form, $\frac{r_1(\cos \theta_1 + i\sin \theta_1)}{r_2(\cos \theta_2 + i\sin \theta_2)}$, simplifies to $r(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))$ where $r = \frac{r_1}{r_2}$. Here, $r_1 = 36$, $r_2 = 6$, $\theta_1 = 2.5$, and $\theta_2 = 1.7$. Thus, $r = \frac{36}{6} = 6$ and $\theta = 2.5 - 1.7 = 0.8$, resulting in $6(\cos 0.8 + i\sin 0.8)$.

Question 11

Given $A = 52 ^ { \circ } , B = 79 ^ { \circ }$, and $a = 5.1$, use the Law of Sines to solve the triangle for the value of $b$. Round answer to two decimal places.

A) $b = 3.92$
B) $b = 4.88$
C) $ b = 5.33$
D) $b = 6.35$
E) $ b = 4.09$

Accepted Answer

The answer of Given \(A = 52 ^ { \circ...

Question 12

Find the projection of $\mathbf { u }$ onto $\mathbf { v }$ if $\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangle$.
A) $\left\langle \frac { 44 } { \sqrt { 13 } } , \frac { 88 } { \sqrt { 13 } } \right\rangle$
B) $\left\langle - \frac { 110 } { \sqrt { 13 } } , \frac { 44 } { \sqrt { 13 } } \right\rangle$
C) $\left\langle - \frac { 110 } { \sqrt { 13 } } , \frac { 88 } { \sqrt { 13 } } \right\rangle$
D) $\left\langle - \frac { 110 } { \sqrt { 13 } } , - \frac { 66 } { \sqrt { 13 } } \right\rangle$
E) $\left\langle \frac { 44 } { \sqrt { 13 } } , - \frac { 66 } { \sqrt { 13 } } \right\rangle$

Accepted Answer

The answer of Find the projection of \(\mathbf { u...

Question 13

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither.
$$\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 30 , - 5 \rangle$$
A) orthogonal
B) parallel
C) neither

Accepted Answer

The answer of Determine whether $\mathbf { u }$ and...

Question 14

Find the trigonometric form of the complex number shown below. $- 6 i$
A) $6 ( \cos 0 + i \sin 0 )$
B) $6 ( \cos \pi + i \sin \pi )$
C) $6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $6 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
E) $6 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Accepted Answer

To find the trigonometric form of a complex number, we need to convert it to the form $a+bi$ and then use the formulas $r=\sqrt{a^2+b^2}$, $\cos	heta=\frac{a}{r}$, and $\sin	heta=\frac{b}{r}$ to find the modulus $r$ and argument $	heta$. Here, we have $-6i=0-6i$, so $a=0$ and $b=-6$. Thus, $r=\sqrt{0^2+(-6)^2}=6$ and $\cos	heta=\frac{0}{6}=0$ while $\sin	heta=\frac{-6}{6}=-1$. Therefore, $	heta=\frac{3\pi}{2}$ since $\cos	heta=0$ and $\sin	heta=-1$ in the third quadrant. Hence, the trigonometric form of $-6i$ is $$6\left(\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}ight)=\boxed{	extbf{(C)}}.$$

Question 15

An airplane is flying at a bearing of $143 ^ { \circ }$ with an airspeed of 335 kilometers per hour. Because of the wind, its ground speed and direction are, respectively, 325 kilometers per hour and $138 ^ { \circ }$. Find the direction and speed of the wind. Round your answer to two decimals.
A) $31.36 ^ { \circ } , 30.47$ kilometers per hour
B) $121.36 ^ { \circ } , 25.71$ kilometers per hour
C) $- 37.01 ^ { \circ } , 30.47$ kilometers per hour
D) $31.36 ^ { \circ } , 25.71$ kilometers per hour
E) $121.36 ^ { \circ } , 25.69$ kilometers per hour

Accepted Answer

To solve this problem, we can use vector addition, where the airplane's velocity vector and the wind's velocity vector add up to the resultant ground velocity vector. Given the airplane's airspeed and bearing, and its ground speed and direction, we can calculate the wind's speed and direction by decomposing the vectors into their north-south and east-west components, and then solving for the wind's components. The correct calculation leads to a wind direction of approximately $31.36^\circ$ and a wind speed of approximately $30.47$ kilometers per hour.

Question 16

Find the component form of vector $\mathbf { v }$.

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

Accepted Answer

The component form of a vector is given by its horizontal and vertical components. Looking at the vector in the image, we see that it moves 4 units to the right (positive x direction) and 5 units down (negative y direction), so the component form is v = ⟨4,−5⟩.

Question 17

In the figure below, $( a , b ) = ( 2,7 )$ and $( c , d ) = ( 7,5 ) .$ Find $( x , y )$ so that $\mathbf { u } = \mathbf { v }$

A) $( 5 , - 2 )$
B) $( 14,7 )$
C) $( 0,3 )$
D) $( 7,14 )$
E) $( 9,12 )$

Accepted Answer

The answer of In the figure below, \(( a ,...

Question 18

Use DeMoivre's Theorem to find the indicated power of the following complex number. $$( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$$
A) 19,683
B) 6,561
C) 2,187
D) 46,656
E) 243

Accepted Answer

DeMoivre's Theorem states that for a complex number in polar form $ r(\cos \theta + i \sin \theta) $, its nth power is $ r^n (\cos(n\theta) + i \sin(n\theta)) $. First, convert $ 3 + 3\sqrt{3}i $ to polar form: the modulus is $ 6 $ and the argument is $ \pi/3 $. Raising to the 6th power gives $ 6^6 (\cos(2\pi) + i \sin(2\pi)) = 46,656 $.

Question 19

Find the vector $\mathbf { v }$ that has a magnitude of 8 and is in the same direction as $\mathbf { u }$, where $\mathbf { u } = \langle - 6 , - 4 \rangle$
A) $\mathbf { v } = \left\langle \frac { - 24 } { \sqrt { 13 } } , \frac { - 16 } { \sqrt { 13 } } \right\rangle$
B) $\mathbf { v } = \left\langle \frac { - 3 } { \sqrt { 13 } } , \frac { - 2 } { \sqrt { 13 } } \right\rangle$
C) $\mathbf { v } = \left\langle \frac { - 8 } { \sqrt { 13 } } , \frac { - 4 } { \sqrt { 13 } } \right\rangle$
D) $\mathbf { v } = \left\langle \frac { - 2 } { \sqrt { 13 } } , \frac { 2 } { \sqrt { 13 } } \right\rangle$
E) $ \mathbf { v } = \left\langle \frac { 4 } { \sqrt { 13 } } , \frac { 6 } { \sqrt { 13 } } \right\rangle$

Accepted Answer

To find a vector with a specific magnitude in the same direction as another vector, you first find the unit vector in the direction of the given vector and then multiply it by the desired magnitude. The magnitude of $\mathbf{u} = \langle -6, -4 \rangle$ is $\sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = \sqrt{4 \cdot 13} = 2\sqrt{13}$. The unit vector in the direction of $\mathbf{u}$ is $\frac{1}{2\sqrt{13}} \langle -6, -4 \rangle = \langle \frac{-6}{2\sqrt{13}}, \frac{-4}{2\sqrt{13}} \rangle = \langle \frac{-3}{\sqrt{13}}, \frac{-2}{\sqrt{13}} \rangle$. To get a vector of magnitude 8 in the same direction, multiply the unit vector by 8: $8 \langle \frac{-3}{\sqrt{13}}, \frac{-2}{\sqrt{13}} \rangle = \langle \frac{-24}{\sqrt{13}}, \frac{-16}{\sqrt{13}} \rangle$.

Question 20

Plot the following complex number.
$3 i$

A)

B)

C)

D)

E)

Accepted Answer

The answer of Plot the following complex number.
$3 i$

A)

B)

C)

D)

E)
...

Quiz 6: Additional Topics in Trigonometry

Let $\mathbf { u } = \langle 6,8 \rangle$ and $\mathbf { v } = \langle 14,10 \rangle$ . Find $\mathbf { u } - \mathbf { v }$

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 50 feet. How much work has been Accomplished? Round answer to one decimal place.

Perform the operation below and leave the result in trigonometric form. $\frac { 4 \left( \cos 58 ^ { \circ } + i \sin 58 ^ { \circ } \right) } { 12 \left( \cos 13 ^ { \circ } + i \sin 13 ^ { \circ } \right) }$

Let $\mathbf { w }$ be a vector with initial point $( 2,1 )$ and terminal point $( 9 , - 3 )$ . Write $\mathbf { w }$ as a linear combination of the standard unit vectors $\mathbf { i }$ and $\mathbf { j }$ .

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number. $( - 6 + 6 i ) ^ { 4 }$

Perform the operation shown below and leave the result in trigonometric form. $\frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }$

Find the projection of $\mathbf { u }$ onto $\mathbf { v }$ if $\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangle$ .

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

Find the trigonometric form of the complex number shown below. $- 6 i$

An airplane is flying at a bearing of $143 ^ { \circ }$ with an airspeed of 335 kilometers per hour. Because of the wind, its ground speed and direction are, respectively, 325 kilometers per hour and $138 ^ { \circ }$ . Find the direction and speed of the wind. Round your answer to two decimals.

Find the component form of vector $\mathbf { v }$ . $Find the component form of vector \mathbf { v } . A) v = \langle - 5,4 \rangle B) v = \langle 4 , - 5 \rangle C) \mathrm { v } = \langle - 4 , - 5 \rangle D) v = \langle 4 , - 3 \rangle E) v = \langle 5,4 \rangle$

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$

Find the vector $\mathbf { v }$ that has a magnitude of 8 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 6 , - 4 \rangle$

Plot the following complex number. $3 i$

Quiz 6: Additional Topics in Trigonometry

Let u=⟨6,8⟩\mathbf { u } = \langle 6,8 \rangleu=⟨6,8⟩ and v=⟨14,10⟩\mathbf { v } = \langle 14,10 \ranglev=⟨14,10⟩ . Find u−v\mathbf { u } - \mathbf { v }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 50 feet. How much work has been Accomplished? Round answer to one decimal place.

In the figure below, a=9,b=11a = 9 , b = 11a=9,b=11 , and c=12c = 12c=12 . Use this information to solve the parallelogram for θ\thetaθ . The diagonals of the parallelogram are represented by ccc and ddd . Round answer to two decimal places.

A plane flies 450 kilometers with a bearing of 319∘319 ^ { \circ }319∘ from City A to City B (see figure below) . The plane then flies 730 kilometers from City B to City C, which is directly west of City A. Find the bearing of the flight from City B to City C. Round your answer to two decimals.

Let w\mathbf { w }w be a vector with initial point (2,1) ( 2,1 ) (2,1) and terminal point (9,−3) ( 9 , - 3 ) (9,−3) . Write w\mathbf { w }w as a linear combination of the standard unit vectors i\mathbf { i }i and j\mathbf { j }j .

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number. (−6+6i) 4( - 6 + 6 i ) ^ { 4 }(−6+6i) 4

Given a=3,b=8a = 3 , b = 8a=3,b=8 , and c=6c = 6c=6 , use the Law of Cosines to solve the triangle for the value of CCC . Round answer to two decimal places.

Perform the operation shown below and leave the result in trigonometric form. 36(cos⁡2.5+isin⁡2.5) 6(cos⁡1.7+isin⁡1.7) \frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }6(cos1.7+isin1.7) 36(cos2.5+isin2.5) ​

Given A=52∘,B=79∘A = 52 ^ { \circ } , B = 79 ^ { \circ }A=52∘,B=79∘ , and a=5.1a = 5.1a=5.1 , use the Law of Sines to solve the triangle for the value of bbb . Round answer to two decimal places.

Find the projection of u\mathbf { u }u onto v\mathbf { v }v if u=⟨5,−4⟩,v=⟨−2,3⟩\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangleu=⟨5,−4⟩,v=⟨−2,3⟩ .

Determine whether u\mathbf { u }u and v\mathbf { v }v are orthogonal, parallel, or neither. u=⟨1,5⟩,v=⟨−30,−5⟩\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 30 , - 5 \rangleu=⟨1,5⟩,v=⟨−30,−5⟩

Find the trigonometric form of the complex number shown below. −6i- 6 i−6i

Find the component form of vector v\mathbf { v }v .

In the figure below, (a,b) =(2,7) ( a , b ) = ( 2,7 ) (a,b) =(2,7) and (c,d) =(7,5) .( c , d ) = ( 7,5 ) .(c,d) =(7,5) . Find (x,y) ( x , y ) (x,y) so that u=v\mathbf { u } = \mathbf { v }u=v

Use DeMoivre's Theorem to find the indicated power of the following complex number. (3+33i) 6( 3 + 3 \sqrt { 3 } i ) ^ { 6 }(3+33​i) 6

Find the vector v\mathbf { v }v that has a magnitude of 8 and is in the same direction as u\mathbf { u }u , where u=⟨−6,−4⟩\mathbf { u } = \langle - 6 , - 4 \rangleu=⟨−6,−4⟩

Plot the following complex number. 3i3 i3i