Question 1

Given that Force 1 = 55 pounds and Force 2 = 130 pounds,find the angle between the forces if the magnitude of the resultant force is 120 pounds.Round your answer to the nearest degree. ​
A)72°
B)59°
C)67°
D)79°
E)64°

Accepted Answer

Let's assume the angle between the forces is θ.
Using the law of cosines, we have:
(120)^2 = (55)^2 + (130)^2 - 2(55)(130)cosθ
cosθ = (55^2 + 130^2 - 120^2) / 2(55)(130)
cosθ = 0.741
θ = cos^-1(0.741) ≈ 67° (rounded to the nearest degree)

Question 2

?If $\mathbf { u } = - 3 \mathbf { i } + 6 \mathbf { j }$ and $\mathbf { v } = - 6 \mathbf { i } + 5 \mathbf { j }$ ,find $\mathbf { w } = 3 \mathbf { u } + 5 \mathbf { v }$ .
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)? $\mathbf { w } = 16 \mathbf { i } - 12 \mathbf { j }$

Accepted Answer

To find $\mathbf { w } = 3 \mathbf { u } + 5 \mathbf { v }$, we first multiply each vector by its scalar: $3\mathbf{u} = 3(-3\mathbf{i} + 6\mathbf{j}) = -9\mathbf{i} + 18\mathbf{j}$ and $5\mathbf{v} = 5(-6\mathbf{i} + 5\mathbf{j}) = -30\mathbf{i} + 25\mathbf{j}$. Adding these results gives $\mathbf{w} = -9\mathbf{i} + 18\mathbf{j} - 30\mathbf{i} + 25\mathbf{j} = -39\mathbf{i} + 43\mathbf{j}$.

Question 3

If $\| \mathbf { u } \| = 4$ and $\| \mathbf { v } \| = 6$ ,and the vectors make angles of 140° and 20° with the x-axis respectively,find the component form of the sum of u and v.Round answers to two decimal places.
A) $\langle 2.57,4.62 \rangle$
B)? $\langle - 0.84,5.22 \rangle$
C)? $\langle - 8.7,0.52 \rangle$
D)? $\langle 8.21 , - 1.01 \rangle$
E)? $\langle - 3.07 , - 5.12 \rangle$

Accepted Answer

To find the component form of the sum of vectors $\mathbf{u}$ and $\mathbf{v}$, calculate each vector's components using their magnitudes and angles, then add them. For $\mathbf{u}$, $\langle 4\cos(140^\circ), 4\sin(140^\circ) \rangle \approx \langle -3.06, 2.57 \rangle$. For $\mathbf{v}$, $\langle 6\cos(20^\circ), 6\sin(20^\circ) \rangle \approx \langle 5.64, 2.05 \rangle$. Adding these gives $\langle -3.06 + 5.64, 2.57 + 2.05 \rangle \approx \langle 2.57, 4.62 \rangle$.

Question 4

Find the component form of v if $\| \mathbf { v } \| = 8$ and the angle it makes with the x-axis is 30°.
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)? $\langle 4 \sqrt { 2 } , 4 \sqrt { 2 } \rangle$

Accepted Answer

The component form of a vector with magnitude $r$ and angle $\theta$ to the x-axis can be found using $r\cos(\theta)$ for the x-component and $r\sin(\theta)$ for the y-component. Given $r = 8$ and $\theta = 30°$, the x-component is $8\cos(30°) = 8(\sqrt{3}/2) = 4\sqrt{3}$ and the y-component is $8\sin(30°) = 8(1/2) = 4$, making the component form $\langle 4\sqrt{3}, 4 \rangle$.

Question 5

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

Accepted Answer

The answer of ?Find the magnitude and direction angle of...

Question 6

Find the magnitude and direction angle of $\mathbf { v } = 6 \left( \cos 80 ^ { \circ } \mathbf { i } + \sin 80 ^ { \circ } \mathbf { j } \right)$ .
A)? $\| \mathbf { v } \| = \sqrt { 6 } ; \theta = 80 ^ { \circ }$
B)? $\| \mathbf { v } \| = 6 ; \theta = 100 ^ { \circ }$
C)? $\| \mathbf { v } \| = 6 ; \theta = 80 ^ { \circ }$
D)? $\| \mathbf { v } \| = 6 ; \theta = 10 ^ { \circ }$
E)? $\| \mathbf { v } \| = \sqrt { 6 } ; \theta = 100 ^ { \circ }$

Accepted Answer

The magnitude of $\mathbf{v}$ is given by the coefficient $6$, as the vector components are already unit vectors in their respective directions. The direction angle $\theta$ is directly given as $80^\circ$, matching the angles used in the cosine and sine functions.

Question 7

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.
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°

Accepted Answer

Using vector addition, we get:

$R_x = 74\cos(170^\circ) + 98\cos(220^\circ) + 105\cos(330^\circ) \approx -83.42$
$R_y = 74\sin(170^\circ) + 98\sin(220^\circ) + 105\sin(330^\circ) \approx -50.12$

$R = \sqrt{R_x^2 + R_y^2} \approx 117.42$ pounds

$	heta = 	an^{-1}\left(\frac{R_y}{R_x}ight) \approx 240.95^\circ$

Therefore, the magnitude of the resultant force is approximately 117.42 pounds and its direction is approximately 240.95° with respect to the positive x-axis. The answer is choice C.

Question 8

Find the component form of v if $\| \mathbf { v } \| = 8$ and the angle it makes with the x-axis is 60°.
A)? $\langle 4 \sqrt { 3 } , 4 \rangle$
B)? $\langle 8,8 \sqrt { 3 } \rangle$
C) $\langle 4,4 \sqrt { 3 } \rangle$
D)? $\langle 8 \sqrt { 3 } , 8 \rangle$
E)? $\langle 4 \sqrt { 2 } , 4 \sqrt { 2 } \rangle$

Accepted Answer

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

Question 9

Find the component form of v if $\| \mathbf { v } \| = 10$ and the angle it makes with the x-axis is 120°.
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)? $\langle - 10,10 \sqrt { 3 } \rangle$

Accepted Answer

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

Question 10

Find the component form of v if $\| \mathbf { v } \| = 6$ and the angle it makes with the x-axis is 150°.
A)? $\langle - 6 \sqrt { 3 } , - 6 \rangle$
B)? $\langle - 6,6 \sqrt { 3 } \rangle$
C)? $\langle - 3 \sqrt { 3 } , 3 \rangle$
D)? $\langle - 3 \sqrt { 2 } , 3 \sqrt { 2 } \rangle$
E)? $\langle - 3,3 \sqrt { 3 } \rangle$

Accepted Answer

To find the component form of v, we need to use the given angle and magnitude. The angle is 150°, which is in the second quadrant. In this quadrant, the x-component is negative and the y-component is positive. 
Using the formula for converting from magnitude and angle to component form, we get:

x-component = magnitude*cos(angle) = 11ea99db_b424*cos(150°) = -ce68
y-component = magnitude*sin(angle) = 11ea99db_b424*sin(150°) = ffa22b157f8

Therefore, the component form of v is 11ea99db_b424_ce68_94ac_59fa22b157f8_TB4590_11. 
The correct answer is C.

Quiz 38:Vectors in the Plane

Given that Force 1 = 55 pounds and Force 2 = 130 pounds,find the angle between the forces if the magnitude of the resultant force is 120 pounds.Round your answer to the nearest degree. ​

?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 .

If ∥u∥=4\| \mathbf { u } \| = 4∥u∥=4 and ∥v∥=6\| \mathbf { v } \| = 6∥v∥=6 ,and the vectors make angles of 140° and 20° with the x-axis respectively,find the component form of the sum of u and v.Round answers to two decimal places.

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 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 magnitude and direction angle of v=6(cos⁡80∘i+sin⁡80∘j) \mathbf { v } = 6 \left( \cos 80 ^ { \circ } \mathbf { i } + \sin 80 ^ { \circ } \mathbf { j } \right) v=6(cos80∘i+sin80∘j) .

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.

Find the component form of v if ∥v∥=8\| \mathbf { v } \| = 8∥v∥=8 and the angle it makes with the x-axis is 60°.

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 if ∥v∥=6\| \mathbf { v } \| = 6∥v∥=6 and the angle it makes with the x-axis is 150°.

Given that Force 1 = 55 pounds and Force 2 = 130 pounds,find the angle between the forces if the magnitude of the resultant force is 120 pounds.Round your answer to the nearest degree.

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

If $\| \mathbf { u } \| = 4$ and $\| \mathbf { v } \| = 6$ ,and the vectors make angles of 140° and 20° with the x-axis respectively,find the component form of the sum of u and v.Round answers to two decimal places.

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

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

Find the magnitude and direction angle of $\mathbf { v } = 6 \left( \cos 80 ^ { \circ } \mathbf { i } + \sin 80 ^ { \circ } \mathbf { j } \right)$ .

Find the component form of v if $\| \mathbf { v } \| = 8$ and the angle it makes with the x-axis is 60°.

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

Find the component form of v if $\| \mathbf { v } \| = 6$ and the angle it makes with the x-axis is 150°.