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Worksheet: Vector Resultants

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:

The displacement vector D i j = ( 3 4 ) 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 1 3 ) i j m
  • B ( 2 1 5 ) i j m
  • C ( 4 1 6 ) i j m
  • D ( 3 1 6 ) i j m
  • E ( 1 1 7 ) i j m

Q3:

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

Q4:

D i j k = ( 2 . 0 0 4 . 0 0 + 1 . 0 0 ) 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?

Q5:

Three dogs pull on a stick, all in different directions, exerting forces F 1 , F 2 , and F 3 . F i j k 1 = ( 1 0 . 0 2 0 . 4 + 2 . 0 ) N, F i k 2 = ( 1 5 . 0 6 . 2 ) N, and F i j 3 = ( 5 . 0 + 1 2 . 5 ) N The forces F 1 , F 2 , and F 3 apply the displacement vector D j k = ( 7 . 9 4 . 2 ) cm to the stick. What work is done by F 3 ?

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

Q6:

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

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

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

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

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

Q7:

Three dogs pull on a stick, all in different directions, exerting forces F 1 , F 2 , and F 3 . F i j k 1 = ( 1 0 . 0 2 0 . 4 + 2 . 0 ) N , F i k 2 = ( 1 5 . 0 6 . 2 ) N , and F i j 3 = ( 5 . 0 + 1 2 . 5 ) N . What is the angle between F 1 and F 2 ?

Q8:

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 .

Q9:

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 .