# Worksheet: Vector Resultants

In this worksheet, we will practice calculating the magnitudes and directions of vectors as well as the magnitudes and directions of sums of vectors.

Q1:

Four dogs named Ang, Bing, Chang, and Dong play a game of tug of war with a toy. Ang pulls on the toy in a direction south of east, Bing in a direction east of north, and Chang in a direction west of north. Ang pulls strongly with a force of magnitude 0.160 kN. Bing pulls even more strongly than Ang with a force of magnitude 0.200 kN, and Chang pulls with a force of magnitude 0.140 kN. When Dong pulls on the toy in such a way that his force is equal in magnitude and opposite in direction to the resultant of the other three forces, the toy does not move in any direction.

What magnitude is the force that Dong pulls the toy with?

At what angle south of west does Dong pull the toy?

Q2:

For the vectors m and, m.

Calculate the magnitude of the vector .

Calculate the magnitude of the vector .

Q3:

Suppose that the vector and the vector .

Calculate the magnitude of the vector .

Calculate the direction angle of the vector .

Calculate the magnitude of the vector .

• A2.00
• B3.00
• C
• D
• E

Calculate the direction angle of the vector .

Q4:

The displacement vector m is added to the displacement vector . The sum of these vectors has a length that is equal to five times the component of . Find .

• A m
• B m
• C m
• D m
• E m

Q5:

For the vectors in the following figure, the positive corresponds to horizontally right and the positive corresponds to vertically upward.

Find the vector that solves the equation .

• A
• B
• C
• D
• E

Find the vector that solves the equation .

• A
• B
• C
• D
• E

Q6:

m is a vector.

What angle does make with the -axis?

What angle does make with the -axis?

What angle does make with the -axis?

Q7:

Three dogs pull on a stick, all in different directions, exerting forces , , and . N, N, and N The forces , , and apply the displacement vector cm to the stick. What work is done by ?

Q8:

Three dogs pull on a stick, all in different directions, exerting forces , , and . , , and . What is the angle between and ?

Q9:

Three displacement vectors , , and have magnitudes 10.0 m, 7.0 m, and 8.0 m respectively. The directions of , , and make counterclockwise angles of , , and , respectively, with the positive -axis, as shown in the diagram.

Calculate .

• A m
• B m
• C m
• D m
• E m

Calculate .

• A m
• B m
• C m
• D m
• E m

Calculate .

• A m
• B m
• C m
• D m
• E m

Q10:

The map given shows the blocks of a city. Each block is a square, with each side being 120 m long. Line shows the path of a tourist walking through the city.

How far does the tourist walk?

What is the magnitude of the displacement from start to finish?

What is the direction, east of north, of the displacement from start to finish?

Q11:

In the diagram, vector has a magnitude of 12.0 m and a direction west of north, and vector has a magnitude of 20.0 m and a direction south of west.

Vector can be defined as . What is the magnitude of vector ?

Vector can be defined as . What is the magnitude of vector ?

Vector can be defined as . What is the magnitude of vector ?

What is the difference between the magnitude of and that of ?

• A26.6 m
• B0 m
• C2.34 m
• D15.4 m
• E53.2 m

What is the difference in direction, in degrees, between and ?

• A
• B
• C
• D
• E

Q12:

The great astronomer Edwin Hubble discovered that all distant galaxies are receding from our Milky Way galaxy with velocities proportional to their distances. It appears to an observer on Earth that we are at the center of an expanding universe. The diagram illustrates this for five galaxies lying along a straight line, with the Milky Way at the center.

Using the data from the figure, what is the velocity of galaxy 1 relative to galaxy 2?

Using the data from the figure, what is the velocity of the Milky Way relative to galaxy 2?

Use the data in the figure to find the average ratio of recession velocity to distance away. Give your answer in units of km/Mly.

These results mean that observers in all galaxies will see themselves at the center of the expanding universe and would likely be aware of relative velocities, concluding that it is not possible to locate the center of expansion with the given information.

If you extrapolate back in time, how long ago would all of the galaxies have been at approximately the same position? There are m in 1 light-year. Use a value of 365 for the number of days in a year. Give your answer in years in standard index form.

• A yr
• B yr
• C yr
• D yr
• E yr