Worksheet: Vectors Applications

In this worksheet, we will practice using vector operations to solve various real-world problems, such as navigation and physics.

Q1:

A police car was moving along a horizontal highway at 47 km/h. It used a radar to measure the speed of a truck moving in the same direction. Given that the reading on the radar was 50 km/h, determine the actual speed of the truck.

Q2:

A truck moved 150 km due east and then 200 km at a direction of 60. Determine the truck’s displacement, giving its magnitude to the nearest kilometre and its direction to the nearest minute.

  • A 304 km, 3 4 4 3 north of east
  • B 180 km, 3 3 4 1 north of west
  • C 200 km, 6 0 north of east
  • D 350 km, 4 9 6 south of east

Q3:

Two cars are moving on the same straight road with velocities of 87 km/h and 85 km/h, respectively. Given that e is a unit vector in the direction of movement of the first car, determine the relative velocity of the first car relative to the second, given that both cars are moving in the same direction.

  • A ( 1 7 2 ) e km/h
  • B ( 8 5 ) e km/h
  • C ( 8 7 ) e km/h
  • D ( 2 ) e km/h

Q4:

Two cars 𝐴 and 𝐵 are moving in opposite directions on the same road at 62 km/h and 31 km/h, respectively. Given that e is a unit vector in the direction of movement of car 𝐴, determine the velocity of car 𝐴 relative to car 𝐵.

  • A ( 9 3 ) e km/h
  • B ( 3 1 ) e km/h
  • C ( 6 2 ) e km/h
  • D ( 3 1 ) e km/h

Q5:

Two children are throwing a ball to each other straight across the back seat of a car, perpendicular to the direction of travel of the car. The ball is being thrown at 10 mph relative to the car, and the car is traveling in a straight line at 25 mph along the road. If one child misses the ball, letting it fly out of the window, find the angle it makes relative to the car’s forward direction, ignoring wind resistance. Give your answer in degrees, correct to one decimal place if necessary.

Q6:

Two children are throwing a ball to each other straight across the back seat of a car, perpendicular to the direction of travel of the car. The ball is being thrown at 8 mph relative to the car, and the car is traveling in a straight line at 45 mph along the road. If one child misses the ball, letting it fly out of the window, find the angle it makes relative to the car’s forward direction ignoring wind resistance. Give your answer in degrees, correct to three decimal places if necessary.

Q7:

A man starts walking from home and walks 3 miles at 20 north of west, then 5 miles at 10 west of south, then 4 miles at 15 north of east. If he walked in a straight line back home, how far would he have to walk and in what direction? Give your answers correct to three decimal places.

  • A 5.351 miles, 56.982 north of west
  • B 5.351 miles, 56.983 west of north
  • C 2.868 miles, 86.474 west of north
  • D 2.868 miles, 86.474 north of west
  • E 9.762 miles, 7.623 north of west

Q8:

Find the values of 𝑘, 𝑚, and 𝑛 that make the two vectors A=𝑘2,𝑚2,4 and B=3,2,𝑛 equal.

  • A 𝑘 = 1 , 𝑚 = 4 , 𝑛 = 1
  • B 𝑘 = 3 , 𝑚 = 2 , 𝑛 = 4
  • C 𝑘 = 5 , 𝑚 = 2 or 2, 𝑛=2 or 2
  • D 𝑘 = 5 , 𝑚 = 2 or 2, 𝑛=2 or 2

Q9:

Velocity is a vector quantity which combines speed and direction. For example, the velocity of a section of the Mississippi river near New Orleans is 3 miles per hour east.

What is the magnitude of this velocity?

What is the direction of this velocity?

  • ANew Orleans
  • B3 miles per hour east
  • Ceast
  • Dmiles per hour east
  • EMississippi

Q10:

Two cars are moving on the same straight road with velocities of 17 km/h and 50 km/h, respectively. Given that e is a unit vector in the direction of movement of the first car, determine the relative velocity of the first car relative to the second, given that both cars are moving in the same direction.

  • A ( 6 7 ) e km/h
  • B ( 5 0 ) e km/h
  • C ( 1 7 ) e km/h
  • D ( 3 3 ) e km/h

Q11:

Two cars 𝐴 and 𝐵 are moving in opposite directions on the same road at 13 km/h and 82 km/h, respectively. Given that e is a unit vector in the direction of movement of car 𝐴, determine the velocity of car 𝐴 relative to car 𝐵.

  • A ( 9 5 ) e km/h
  • B ( 6 9 ) e km/h
  • C ( 1 3 ) e km/h
  • D ( 6 9 ) e km/h

Q12:

A police car was moving along a horizontal highway at 33 km/h. It used a radar to measure the speed of a truck moving in the same direction. Given that the reading on the radar was 100 km/h, determine the actual speed of the truck.

Q13:

Given that vi=17 and vi=8, find v.

  • A 9 i
  • B 2 5 i
  • C 9 i
  • D 2 5 i

Q14:

The forces Fij=107, Fij=𝑎, and Fij=5+(𝑏10) act on a particle, where i and j are two perpendicular unit vectors. Given that the forces’ resultant Rij=133, determine the values of 𝑎 and 𝑏.

  • A 𝑎 = 2 8 , 𝑏 = 5
  • B 𝑎 = 2 , 𝑏 = 1 5
  • C 𝑎 = 8 , 𝑏 = 1 5
  • D 𝑎 = 8 , 𝑏 = 5
  • E 𝑎 = 8 , 𝑏 = 5

Q15:

Given that Fij=85, Fij=155, and their resultant Rij=𝑎𝑏, determine the values of 𝑎 and 𝑏.

  • A 𝑎 = 7 , 𝑏 = 0
  • B 𝑎 = 7 , 𝑏 = 1 0
  • C 𝑎 = 7 , 𝑏 = 1 0
  • D 𝑎 = 7 , 𝑏 = 1 0
  • E 𝑎 = 2 3 , 𝑏 = 1 0

Q16:

If vi=76 and vi=5, find v.

  • A 8 1 i
  • B 7 1 i
  • C 7 1 i
  • D 8 1 i

Q17:

A river is one half mile wide with a current flowing at 2 miles per hour from east to west. A man can swim at 3 miles per hour in still water. He stands on the south bank, so how many degrees east of north should he head at this rate in order to travel directly north across the river? Give your answer correct to one decimal place. What would the answer to this problem be if the river flowed at 3 miles per hour and the man could swim only at the rate of 2 miles per hour?

  • A 4 1 . 8 , 4 2 . 0
  • B 6 3 . 4 , 4 2 . 0
  • C 3 3 . 7 , it is not possible for him to head directly north in the second case.
  • D 4 1 . 8 , it is not possible for him to head directly north in the second case.
  • E 6 3 . 4 , it is not possible for him to head directly north in the second case.

Q18:

A woman leaves home and walks 3 miles west and then 2 miles southwest. How far from home is she? In what direction must she walk to directly head home? Give the distance, in miles, correct to two decimal places and the direction, in degrees, to one decimal place.

  • A 4.15 miles, 1 4 . 3 north of east
  • B 4.64 miles, 1 7 . 8 north of east
  • C 4.64 miles, 2 7 . 2 east of north
  • D 4.15 miles, 3 0 . 7 north of east
  • E 21.50 miles, 2 7 . 2 north of east

Q19:

A man started walking from home and walked 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far did he walk in total? If he walked in a straight line back home, how far would he have to walk? Give your answer correct to three decimal places if necessary.

  • A 17 miles, 13.8 miles
  • B 15 miles, 10.3 miles
  • C 17 miles, 10.3 miles
  • D 13 miles, 6.7 miles
  • E 15 miles, 13.8 miles

Q20:

A plane is heading north with an airspeed of 500 km/h. However, there is a wind blowing from the southeast toward northwest at 50 km/h. How many degrees off course will the plane end up flying? What is the plane’s speed relative to the ground? Give your answers correct to one decimal place.

  • A 8 6 . 2 , 536.5 km/h
  • B 8 6 . 2 , 552.3 km/h
  • C 1 . 5 , 550.0 km/h
  • D 3 . 8 , 552.3 km/h
  • E 3 . 8 , 536.5 km/h

Q21:

A body moved 28 m due east and then 14 m due north. Determine the body’s displacement, stating its direction to the nearest minute.

  • A 1 4 5 m, 6 3 2 6 north of east
  • B 1 4 5 m, 2 6 3 4 north of east
  • C 1 4 3 m, 2 6 3 4 north of east
  • D 1 4 3 m, 6 3 2 6 north of east

Q22:

An airplane was flying due north at 150 miles per hour . A wind was pushing the airplane due east at 40 miles per hour. After 1 hour, the plane started flying 30 east of north. Assuming the plane started at (0,0), find, in ordered pair, the position of the plane after 2 hours from starting the flight. Let north be the direction of the positive 𝑦-axis and east the positive 𝑥-axis.

  • A ( 2 3 0 , 3 0 0 )
  • B 8 0 + 7 5 3 , 2 2 5
  • C 8 0 + 7 5 3 , 1 5 0 + 7 5 3
  • D ( 1 5 5 , 2 2 5 )
  • E 1 5 5 , 1 5 0 + 7 5 3

Q23:

An airplane is heading north at an airspeed of 600 km/h, but there is a wind blowing from the southwest at 80 km/h. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground? Give your answers correct to one decimal place.

  • A 8 5 . 1 , 684.7 km/h
  • B 4 . 9 , 659.0 km/h
  • C 2 . 5 , 680.0 km/h
  • D 4 . 9 , 684.7 km/h
  • E 8 5 . 1 , 659.0 km/h

Q24:

An airplane flies at an airspeed of 500 km/h. If there is a wind blowing at 80 km/h from northwest, on what bearing must the pilot fly the plane to travel due north? Give your answer to three decimal places.

Q25:

Two points 𝐴 and 𝐵 lie on a field. James wants to fly his drone from 𝐴 to 𝐵 and measures the bearing to be 020. He also measures that a 10 mph wind is blowing on a bearing of 300.

If he is going to fly the drone at 25 mph, on what bearing would he have to fly to travel from 𝐴 to 𝐵? Give your answer to two decimal places.

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

Calculate the ground speed of the remote control plane to two decimal places.

  • A19.89 mph
  • B24.71 mph
  • C74.15 mph
  • D12.14 mph
  • E20.12 mph

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