Question 1

Vectors 
, 
,and 
Have magnitudes 6,11,and 22.When these vectors are added,what is the least possible magnitude of their addition?
A)25
B)15
C)5
D)3

Accepted Answer

The triangle inequality states that the magnitude of the sum of two vectors is always greater than or equal to the difference of their magnitudes. Therefore, in order to minimize the magnitude of the sum of the three given vectors, we should try to make the sum of any two vectors as close to the magnitude of the third vector as possible.

If we add vectors (a) and (b), we get a magnitude of 17, which is closer to the magnitude of vector (c) than the magnitudes of (a) and (b) individually. Therefore, the minimum magnitude of the sum of the three vectors is 22 - 17 = 5, which is choice C.

Question 2

Of vectors with lengths 5 m,3 m,and 4 m,which has the greatest magnitude x component?
A)the one with length 5 m
B)the one with length 3 m
C)the one with length 4 m
D)It could be any of the three.

Accepted Answer

The magnitude of the x-component of a vector depends on both its length and its direction. Without information on the direction or angle of the vectors relative to the x-axis, any of the vectors could have the greatest magnitude x-component.

Question 3

Vector 1 is 7 units long and is at 70°.Vector 2 is 5 units long and is at 155°.Vector 3 is 3 units long and is at 225°.Which vector has equal-magnitude components?
A)Vector 1
B)Vector 2
C)Vector 3
D)None of the vectors has equal-magnitude components.

Accepted Answer

To determine if a vector has equal-magnitude components, we need to see if its horizontal and vertical components are equal.

The horizontal component of Vector 1 is 7*cos(70°) ≈ 2.2799. 
The vertical component of Vector 1 is 7*sin(70°) ≈ 6.5792. 
So, the components are not equal.

The horizontal component of Vector 2 is 5*cos(155°) ≈ -1.9533. 
The vertical component of Vector 2 is 5*sin(155°) ≈ 4.6712. 
So, the components are not equal.

The horizontal component of Vector 3 is 3*cos(225°) ≈ -2.1213. 
The vertical component of Vector 3 is 3*sin(225°) ≈ -2.1213. 
So, the components are equal.

Therefore, the answer is choice C, Vector 3.

Question 4

I walk six miles in a straight line in a direction north of east,and I end up four miles north and several miles east.How many degrees north of east have I walked?
A)19°
B)42°
C)60°
D)71°

Accepted Answer

We can draw a right-angled triangle where the hypotenuse is the distance covered by the person, which is 6 miles, and the vertical side represents the distance covered northwards, which is 4 miles.
Using Pythagoras theorem, we can find the length of the horizontal side: 
Horizontal side = sqrt(6^2 - 4^2) = 2√5 miles
The angle between the hypotenuse and vertical side represents the direction north of east.
tan θ = (opposite/adjacent) = (4/2√5) = 2/√5
θ = tan⁻¹(2/√5) = 41.81°
Therefore, the person walked 41.81° north of east, which is closest to 42° as given in option B.

Question 5

A taxicab moves five blocks due north,five blocks due east,and another three blocks due north.Assume all blocks are of equal size.How far is the taxi from its starting point?
A)13 blocks
B)9.4 blocks
C)9.2 blocks
D)8.6 blocks

Accepted Answer

The taxi's total movement north is 8 blocks (5 + 3) and east is 5 blocks. Using the Pythagorean theorem (since the path forms a right triangle), the distance from the starting point is √(8^2 + 5^2) = √(64 + 25) = √89 ≈ 9.4 blocks.

Question 6

Vector 
Is 3.00 units in length and points along the positive x axis;vector 
Is 5.00 units in length and points along a direction 150° from the positive x axis.What is the direction of their sum with respect to the positive x axis?
A)62.0°
B)13.0°
C)118°
D)103°

Accepted Answer

We can add the two vectors graphically by placing the tail of the second vector at the head of the first vector and then drawing the resultant vector from the tail of the first vector to the head of the second vector.

Alternatively, we can add the vectors using their components. The first vector has components (3,0) and the second vector has components (5cos150°,5sin150°) = (-2.55, -4.33).

The sum of the two vectors can be found by adding their corresponding components:

(3,0) + (-2.55, -4.33) = (0.45, -4.33)

The direction of the resultant vector with respect to the positive x axis can be found using trigonometry:

θ = tan⁻¹(-4.33/0.45) ≈ 118°

Therefore, the answer is C, 118°.

Question 7

A student adds two vectors with magnitudes of 200 and 70.Which one of the following is the only possible choice for the magnitude of their sum?
A)100
B)280
C)200
D)70

Accepted Answer

The magnitude of the sum of two vectors can range from the absolute difference to the sum of their magnitudes. Here, the range is from 130 (200-70) to 270 (200+70). 200 is within this range.

Question 8

Vector 
Is 5 m long and vector 
Is 12 m long.The length of the sum of the vectors must be:
A)some value from 7 m to 17 m.
B)13 m.
C)17 m.
D)8 m.

Accepted Answer

The answer of Vector 
Is 5 m long and vector...

Question 9

Doubling the radius of a sphere results in increasing its volume by a factor of
A)2.
B)4.
C)8.
D)8π.

Accepted Answer

The volume of a sphere is given by V = (4/3)πr^3. When the radius (r) is doubled, it becomes 2r, and the new volume is V' = (4/3)π(2r)^3 = (4/3)π(8r^3) = 8V. Therefore, the volume is increased by a factor of 8, which is option C.

Question 10

Two ships set sail from the same port.Ship A sails a distance of 35.0 kilometers in a direction 30.0° north of east,and drops anchor.Ship B sails due south.When ship B is 30.0 km from the port,how far apart are the ships?
A)​46.1 km
B)​56.3 km
C)​65.0 km
D)​75.9 km

Accepted Answer

The answer of Two ships set sail from the same...

Question 11

Note the expression: y = x².Which statement is most consistent with this expression? A)If y doubles,then x quadruples. B)y is greater than x. C)If x doubles,then y doubles. D)If x doubles,then y quadruples.

Accepted Answer

As per the given expression, y = x². On simplifying it, we get y = x*2^1*1.2^1*1*1. Hence, if x doubles, i.e., x*2, then the value of the expression becomes y = (x*2)*2^1*1.2^1*1*1 = x*2^2*1.2^1*1*1 = x quadruples. Therefore, option D is the correct choice.

Question 12

​An airplane takes off due west at an angle of 16.0° with the horizontal.When the airplane is at a vertical height of 400 ft,what is its displacement vector? Let east be the +x direction and +z be upwards.
A)​(385  + 400  )ft
B)​(-110  - 400  )ft
C)​(717  - 400  )ft
D)​(-1 390  + 400  )ft

Accepted Answer

The answer of ​An airplane takes off due west at...

Question 13

Note the expression: y = A/x³.Which statement is most consistent with this expression? A)y is less than A. B)If x is halved,y is multiplied by eight. C)If x is doubled,y is multiplied by a factor of 8. D)y is greater than x.

Accepted Answer

If x is halved, then x/2 is substituted into the expression, making the denominator 2S1U1P13S1S1P0. Since the denominator is in the exponent, we can use exponent rules to simplify it to S1U1P13S1S1P0 - 1 = S1U1P03S1S1P0. This means the denominator has been multiplied by a factor of 8, so y is multiplied by 1/8. Therefore, if x is halved, y is multiplied by eight.

Question 14

Terrence goes for a walk.He walks 2.0 km north,then turns and walks 4.0 km south of east.What is his displacement vector? Take east to be in the + 
-direction and north to be in the + 
-direction.
A)(2.0  + 4.0  )km
B)(2.8  - 0.8  )km
C)(4.8  + 5.7  )km
D)(5.7  - 2.8  )km

Accepted Answer

The displacement vector is the straight line distance between the starting point and ending point. We can use Pythagoras theorem to find the magnitude of the displacement vector and we can use trigonometry to find its direction.

First, we can find the horizontal and vertical components of the displacement vector using the given distances:
Horizontal component = 4.0 km cos(45°) = 2.8 km
Vertical component = 2.0 km + 4.0 km sin(45°) = 4.8 km

Using Pythagoras theorem, the magnitude of the displacement vector is:
Magnitude = √(2.8 km)² + (4.8 km)² = 5.6 km

Using trigonometry, the direction of the displacement vector is:
Direction = tan⁻¹(4.8 km / 2.8 km) = 59° north of east

Therefore, the displacement vector is (2.8 11ea800b_18e7_c80e_846e_5752c8ea965c_TB6864_11 - 0.8 11ea800b_18e7_c80f_846e_0942631018e6_TB6864_11) km, which corresponds to Choice B.

Question 15

Vector 
Is 2 m long,and vector 
Is 5 m long.The length of the sum of the vectors must be
​
A)6 m.
B)7 m.
C)13 m.
D)some value from 3 m to 7 m.
E)​3 m.

Accepted Answer

The answer of Vector 
Is 2 m long,and vector 
Is...

Question 16

Vector 
Is 3.0 units in length and points along the positive x axis;vector 
Is 5.0 units in length and points along a direction 150° from the positive x axis.What is the magnitude of the resulting vector when vectors 
And 
Are added?
A)7.0
B)6.7
C)2.8
D)2.1

Accepted Answer

The answer of Vector 
Is 3.0 units in length and...

Question 17

Arvin the Ant travels 30 cm eastward,then 25 cm northward,and finally 10 cm westward.In what direction is Arvin now in with respect to his starting point?
A)59° N of E
B)29° N of E
C)29° N of W
D)51° N of E

Accepted Answer

The answer of Arvin the Ant travels 30 cm eastward,then...

Question 18

A circle has an area of 3.0 m².A second circle has double the radius of the first.The area of the second circle is ____ times that of the first. A)0.8 B)2.0 C)4.0 D)6.0

Accepted Answer

We can start by using the formula for the area of a circle:

A = πr^2

For the first circle, we know that:

πr₁² = 3.0 m

For the second circle, we know that the radius is twice as large as the first circle:

r₂ = 2r₁

We can use this to find the area of the second circle:

πr₂² = π(2r₁)² = 4πr₁²

So the area of the second circle is 4 times the area of the first circle:

4πr₁² / πr₁² = 4

Therefore, the answer is C) 4.0.

Question 19

Find the sum of the following two vectors: (i)50 units due east and (ii)100 units 30° south of west.
A)100 units 30° south of west
B)62 units 54° south of west
C)87 units 60° south of west
D)62 units 15° south of west

Accepted Answer

To find the sum of the two vectors, we first break them into their components. The first vector is 50 units due east, so its components are (50, 0). The second vector is 100 units 30° south of west, which means its components can be found using trigonometry: its x-component (westward) is $100 \cos(30^\circ)$ and its y-component (southward) is $100 \sin(30^\circ)$. Thus, its components are $(-100 \cos(30^\circ), -100 \sin(30^\circ))$ or approximately $(-86.6, -50)$. Adding the components of the two vectors gives us $(50 - 86.6, 0 - 50) = (-36.6, -50)$. The magnitude of the resultant vector is $\sqrt{(-36.6)^2 + (-50)^2} \approx 62$ units, and the direction can be found using the tangent function: $\tan^{-1}\left(\frac{-50}{-36.6}\right) \approx 54^\circ$ south of west.

Question 20

Vector ​A is 2.0 
​ + 3.5 
)Vector B is -3.1 
- 3)1 
)What is A + 2B?
A)​-4.2  + -2.7 
B)​-1.1  - 0.4 
C)​5.1  + 6.6 
D)​8.2  + 9.7

Accepted Answer

To find A + 2B, we need to add the corresponding components of A and 2B. The x-component of A is 2.0, the x-component of 2B is 2*(-3.1) = -6.2, so the x-component of A+2B is 2.0 + (-6.2) = -4.2. Similarly, the y-component of A is 3.5, and the y-component of 2B is 2*(-1) = -2, so the y-component of A+2B is 3.5 + (-2) = 1.5. Therefore, A+2B is -4.2 TB6864_11 + 1.5 TB6864_11, which corresponds to choice A.

Quiz 1: Units, Trigonometry, and Vectors

Vectors , ,and Have magnitudes 6,11,and 22.When these vectors are added,what is the least possible magnitude of their addition?

Of vectors with lengths 5 m,3 m,and 4 m,which has the greatest magnitude x component?

Vector 1 is 7 units long and is at 70°.Vector 2 is 5 units long and is at 155°.Vector 3 is 3 units long and is at 225°.Which vector has equal-magnitude components?

I walk six miles in a straight line in a direction north of east,and I end up four miles north and several miles east.How many degrees north of east have I walked?

A taxicab moves five blocks due north,five blocks due east,and another three blocks due north.Assume all blocks are of equal size.How far is the taxi from its starting point?

Vector Is 3.00 units in length and points along the positive x axis;vector Is 5.00 units in length and points along a direction 150° from the positive x axis.What is the direction of their sum with respect to the positive x axis?

A student adds two vectors with magnitudes of 200 and 70.Which one of the following is the only possible choice for the magnitude of their sum?

Vector Is 5 m long and vector Is 12 m long.The length of the sum of the vectors must be:

Doubling the radius of a sphere results in increasing its volume by a factor of

Two ships set sail from the same port.Ship A sails a distance of 35.0 kilometers in a direction 30.0° north of east,and drops anchor.Ship B sails due south.When ship B is 30.0 km from the port,how far apart are the ships?

Note the expression: y = x².Which statement is most consistent with this expression?

An airplane takes off due west at an angle of 16.0° with the horizontal.When the airplane is at a vertical height of 400 ft,what is its displacement vector? Let east be the +x direction and +z be upwards.

Note the expression: y = A/x³.Which statement is most consistent with this expression?

Terrence goes for a walk.He walks 2.0 km north,then turns and walks 4.0 km south of east.What is his displacement vector? Take east to be in the + -direction and north to be in the + -direction.

Vector Is 2 m long,and vector Is 5 m long.The length of the sum of the vectors must be

Vector Is 3.0 units in length and points along the positive x axis;vector Is 5.0 units in length and points along a direction 150° from the positive x axis.What is the magnitude of the resulting vector when vectors And Are added?

Arvin the Ant travels 30 cm eastward,then 25 cm northward,and finally 10 cm westward.In what direction is Arvin now in with respect to his starting point?

A circle has an area of 3.0 m².A second circle has double the radius of the first.The area of the second circle is ____ times that of the first.

Find the sum of the following two vectors: (i) 50 units due east and (ii) 100 units 30° south of west.

Vector A is 2.0 + 3.5 ) Vector B is -3.1 - 3) 1 ) What is A + 2B?