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 55.0 south of east, Bing in a direction 60.0 east of north, and Chang in a direction 55.0 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 Aijk=(3.004.00+4.00) m and, Bijk=(2.00+3.007.00) m.

Calculate the magnitude of the vector AB+.

Calculate the magnitude of the vector AB.

Q3:

Suppose that the vector Aij=3.002.00 and the vector Bij=1.004.00.

Calculate the magnitude of the vector AB+.

Calculate the direction angle of the vector AB+.

Calculate the magnitude of the vector AB.

  • A2.00
  • B3.00
  • C 2
  • D 3 . 0 0 2
  • E 2 . 0 0 2

Calculate the direction angle of the vector AB.

Q4:

The displacement vector Dij=(34) m is added to the displacement vector R. The sum of these vectors (+)DR has a length that is equal to five times the j component of D. Find R.

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

Find the vector R that solves the equation DRF+=.

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

Find the vector R that solves the equation CDRF2+5=3.

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

Q6:

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?

Q7:

Three dogs pull on a stick, all in different directions, exerting forces F, F, and F. Fijk=(10.020.4+2.0) N, Fik=(15.06.2) N, and Fij=(5.0+12.5) N The forces F, F, and F apply the displacement vector Djk=(7.94.2) cm to the stick. What work is done by F?

  • A 1 . 0 J
  • B 1 . 5 J
  • C1.5 J
  • D 0 . 5 5 J
  • E1.0 J

Q8:

Three dogs pull on a stick, all in different directions, exerting forces F, F, and F. Fijk=(10.020.4+2.0)N, Fik=(15.06.2)N, and Fij=(5.0+12.5)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 35, 110, and 30, respectively, with the positive 𝑥-axis, as shown in the diagram.

Calculate ABC++.

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

Calculate AB.

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

Calculate ABC3+.

  • A ( 2 0 . 5 2 9 . 0 ) i j m
  • B ( 2 2 . 2 + 1 9 . 3 ) i j m
  • C ( 1 9 . 3 + 2 5 . 9 ) i j m
  • D ( 2 3 . 3 + 2 1 . 9 ) i j m
  • E ( 2 2 . 3 + 2 9 . 5 ) 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 20.0 west of north, and vector B has a magnitude of 20.0 m and a direction 40.0 south of west.

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

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

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

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

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

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

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

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.