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 5 5 . 0 south of east, Bing in a direction 6 0 . 0 east of north, and Chang in a direction 5 5 . 0 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 A i j k = ( 3 . 0 0 4 . 0 0 + 4 . 0 0 ) m and, B i j k = ( 2 . 0 0 + 3 . 0 0 7 . 0 0 ) m.

Calculate the magnitude of the vector A B + .

Calculate the magnitude of the vector A B .

Q3:

Suppose that the vector A i j = 3 . 0 0 2 . 0 0 and the vector B i j = 1 . 0 0 4 . 0 0 .

Calculate the magnitude of the vector A B + .

Calculate the direction angle of the vector A B + .

Calculate the magnitude of the vector A B .

  • A2.00
  • B 3 . 0 0 2
  • C 2 . 0 0 2
  • D3.00
  • E 2

Calculate the direction angle of the vector A B .

Q4:

The displacement vector 𝐷 = 3 𝑖 4 𝑗 m is added to the displacement vector 𝑅 . The sum of these vectors ( 𝐷 + 𝑅 ) has a length that is equal to five times the 𝑗 component of 𝐷 . Find 𝑅 .

  • A ( 3 𝑖 1 3 𝑗 ) m
  • B ( 2 𝑖 1 5 𝑗 ) m
  • C ( 4 𝑖 1 6 𝑗 ) m
  • D ( 3 𝑖 1 6 𝑗 ) m
  • E ( 1 𝑖 1 7 𝑗 ) m

Q5:

For the vectors in the following figure, the positive @ 𝑥 -axis corresponds to horizontally right and the positive @ 𝑦 -axis corresponds to vertically upwards.

Find the vector @ 𝑅 that solves the equation @ 𝐷 + @ 𝑅 = @ 𝐹 .

  • A 1 . 5 6 @ 𝑖 1 6 . 6 3 @ 𝑗
  • B 1 . 0 3 @ 𝑖 1 8 . 9 @ 𝑗
  • C 0 . 8 1 @ 𝑖 2 6 . 1 @ 𝑗
  • D 1 . 3 5 @ 𝑖 2 2 . 0 @ 𝑗
  • E 1 . 1 1 @ 𝑖 1 4 . 5 @ 𝑗

Find the vector @ 𝑅 that solves the equation @ 𝐶 2 @ 𝐷 + 5 @ 𝑅 = 3 @ 𝐹 .

  • A 1 8 . 0 @ 𝑖 + 2 . 2 4 @ 𝑗
  • B 1 6 . 3 @ 𝑖 + 0 . 1 4 @ 𝑗
  • C 1 4 . 9 @ 𝑖 + 1 . 7 5 @ 𝑗
  • D 1 9 . 1 @ 𝑖 + 2 . 0 2 @ 𝑗
  • E 2 0 . 3 @ 𝑖 + 0 . 7 9 @ 𝑗

Q6:

𝐷 = ( 2 . 0 0 𝑖 4 . 0 0 𝑗 + 1 . 0 0 𝑘 ) m is a vector.

What angle does 𝐷 make with the 𝑥 -axis?

What angle does 𝐷 make with the 𝑦 -axis?

What angle does 𝐷 make with the 𝑧 -axis?

Q7:

Three dogs pull on a stick, all in different directions, exerting forces 𝐹 , 𝐹 , and 𝐹 . 𝐹 = ( 1 0 . 0 𝑖 2 0 . 4 𝑗 + 2 . 0 𝑘 ) N, 𝐹 = ( 1 5 . 0 𝑖 6 . 2 𝑘 ) N, and 𝐹 = ( 5 . 0 𝑖 + 1 2 . 5 𝑗 ) N The forces 𝐹 , 𝐹 , and 𝐹 apply the displacement vector 𝐷 = ( 7 . 9 𝑗 4 . 2 𝑘 ) cm to the stick. What work is done by 𝐹 ?

  • A 1.0 J
  • B 1 . 5 J
  • C 1.5 J
  • D 1 . 0 J
  • E 0 . 5 5 J

Q8:

Three dogs pull on a stick, all in different directions, exerting forces 𝐹 , 𝐹 , and 𝐹 . 𝐹 = ( 1 0 . 0 𝑖 2 0 . 4 𝑗 + 2 . 0 𝑘 ) N , 𝐹 = ( 1 5 . 0 𝑖 6 . 2 𝑘 ) N , and 𝐹 = ( 5 . 0 𝑖 + 1 2 . 5 𝑗 ) N . What is the angle between 𝐹 and 𝐹 ?

Q9:

Three displacement vectors A , B , and C have magnitudes 10.0 m, 7.0 m, and 8.0 m respectively. The directions of A , B , and C make counterclockwise angles of 3 5 , 1 1 0 , and 3 0 , respectively, with the positive 𝑥 -axis, as shown in the diagram.

Calculate A B C + + .

  • A ( 8 . 3 3 . 2 ) i j m
  • B ( 8 . 3 + 3 . 7 ) i j m
  • C ( 1 2 . 0 + 8 . 0 0 ) i j m
  • D ( 1 2 . 7 + 3 . 2 ) i j m
  • E ( 1 3 . 1 2 . 8 ) i j m

Calculate A B .

  • A ( 1 0 . 6 + 1 2 . 3 ) i j m
  • B ( 1 0 . 6 + 1 3 . 2 ) i j m
  • C ( 1 6 . 0 1 2 . 3 ) i j m
  • D ( 6 . 1 + 1 0 . 3 ) i j m
  • E ( 1 1 . 6 + 1 1 . 3 ) i j m

Calculate A B C 3 + .

  • A ( 1 9 . 3 + 2 5 . 9 ) i j m
  • B ( 2 2 . 2 + 1 9 . 3 ) i j m
  • C ( 2 2 . 3 + 2 9 . 5 ) i j m
  • D ( 2 3 . 3 + 2 1 . 9 ) i j m
  • E ( 2 0 . 5 2 9 . 0 ) i j m

Q10:

The map given shows the blocks of a city. Each block is a square, with each side being 120 m long. Line 𝐶 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 𝐴 has a magnitude of 12.0 m and a direction 2 0 . 0 west of north, and vector 𝐵 has a magnitude of 20.0 m and a direction 4 0 . 0 south of west.

Vector 𝑅 can be defined as 𝑅 = 𝐴 + 𝐵 . What is the magnitude of vector 𝑅 ?

Vector 𝑅 can be defined as 𝑅 = 𝐴 𝐵 . What is the magnitude of vector 𝑅 ?

Vector 𝑅 can be defined as 𝑅 = 𝐵 𝐴 . What is the magnitude of vector 𝑅 ?

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

  • A 0 m
  • B 15.4 m
  • C 2.34 m
  • D 53.2 m
  • E 26.6 m

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

  • A 4 . 6 6
  • B 9 0 . 0
  • C 0 . 0 0
  • D 1 8 0
  • E 9 4 . 6 6

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