Vector Practice Problems
1. Write a vector for each of the following diagrams in terms of its components. Write your answer in
𝒗 = 𝑎𝒊 + 𝑏𝒋 form.
a)
b) A vector with a magnitude of 60
and a bearing of 𝑆19°𝑊.
𝑣 = 603.77𝒊 − 524.85𝒋
𝑣 = −19.53𝒊 − 56.73𝒋
c) A vector with a length of 5
and a bearing of 𝑁81°𝑊.
𝑣 = −4.94𝒊 + 0.78𝒋
2. Let 𝑟 = 7, 𝑠 = 2, 𝒖 = −1, 4 , and 𝒗 = 5, − 2 . Find 𝑠𝒖 + 𝑟𝒗 .
15 5 ≈ 33.541
3. Find the unit vector in the direction of 3, 5 .
? ?@ A ?@
?@
,
?@
4. Let 𝐴 = −2, 3 and 𝐵 = 4, 5 . Express the vector 𝐴𝐵 in 𝑣 = 𝑎𝒊 + 𝑏𝒋 form.
6𝒊 + 2𝒋
5. Find 𝒖 ∙ 𝒗 given the angle between 𝒖 and 𝒗 is 𝜃 = 55° and 𝒖 = 3 and 𝒗 = 2.
6𝑐𝑜𝑠55° ≈ 3.441
6. Let 𝒂 = 6, 2 and 𝒃 = −1, 3 . Find 𝒑𝒓𝒐𝒋𝒂 𝒃.
0, 0
7. Find the angle between 𝒖 = −2, 5 , and 𝒗 = 1, −3 .
𝜃 = 176.63°
?M
8. Find the vector with magnitude 5 and direction 𝜃 = . Write your answer in 𝑣 = 𝑎𝒊 + 𝑏𝒋 form.
−
A N
N
𝒊+
A N
N
@
𝒋
9. Let 𝒖 = 1, 1 . Find the vector 𝒗 such that 𝒖 ∙ 𝒗 = 𝟖 and 𝒗 = 32.
4, 4
10. Find the component form of the vector originating from 3, −1 with terminal point at −5, 6 .
−8, 7
11. Add the vectors to determine the resultant vector; 𝒗 has a magnitude of 3.0 m/s and a direction of 45°
and 𝒘 has a magnitude of 5.0 m/s and a direction of 135°.
(5.83 m/s, 104°)
12. What does the word “magnitude” mean?
13. What does the word “scalar” mean?
Word Problems:
Speed and direction:
1. A plane travels due south at 485 km/h. A 38 km/h wind blows 𝑆41°𝐸. Find the ground speed and the
direction of the plane.
(Speed is 514.28 km/h. Direction is 𝑆2.77°𝐸)
2. An airplane is travelling 450 mph heading 𝑁60°𝐸. The wind is blowing form the north at 42 mph. find
the ground speed and the direction of the airplane.
(Speed is 430.54 mph. Direction is 𝑁64.84°𝐸)
3. A plane is heading 𝑆52°𝑊 at 310 mph. the wind is blowing at 𝑆78°𝐸 at 24 mph. Find the ground speed
and the direction of the plane.
(Speed 295.15 mph. Direction is 𝑆48.43°𝑊)
4. A boat travels 30 mph due west. If there is a 7mph current at 𝑁30°𝑊, find the actual speed and
direction of the boat.
(Speed 34.04 mph. Direction is 𝑁79.74°𝑊)
5. An airplane’s velocity with no wind is 580 km/h with a bearing of 𝑁60°𝐸. The wind at the altitude of
the plane has a velocity of 60 km/h and is blowing southeast (which means 𝑆45°𝐸). What is the true
speed and bearing of the plane?
(Speed 598.34 km/h. Direction is 𝑁65.56°𝐸)
6. A river is flowing at 2.4 mph when a girl rows across it. If the girl rows at a still-water speed of 3.1 mph
and heads the boat perpendicular to the direction of the current, find the resultant speed of the boat.
If the river is 1 mile across, how far down the river will she have moved by the time she reaches the far
side of the river? How long will it take to cross the river?
(Speed 3.92 mph, Distance 0.77 miles, Time 0.32 hours or 19.35 minutes)
7. At what bearing should a pilot head, if she wants to fly due south at 520 mph when a 40 mph wind is
blowing form the east?
(Speed 521.54 mph, 𝑆4.40°𝐸)
8. An airplane is flying with a bearing of 𝑁35°𝐸 at 550 mph. A wind is blowing with a bearing of 𝑁45°𝐸 at
30 mph. Find the resultant speed and direction of the plane.
(Speed 579.57 mph. Direction is 𝑁35.52°𝐸)
9. A plane is flying with a bearing of 𝑁55°𝐸 at 500 mph. A wind is blowing with a bearing of 𝑁65°𝐸 at 35
mph. Find the resultant speed and direction of the plane.
(Speed 534.50 mph. Direction is 𝑁55.65°𝐸)
10. A 50-meter-wide river is flowing at 5 m/sec. A person in a kayak rows at a speed of 3 m/sec. What
would be the maximum possible speed of the kayaker and in what direction should they point? What
would be the slowest possible speed of the kayaker and in what direction should they point? If the
kayaker tries to kayak straight across the river (perpendicular to the current), how long would it take
for the kayaker to cross?
(Max. speed 8 m/sec – in direction of current, min. speed 2 m/sec – in direction of current,
16.7 seconds)
11. A Cyclist heads due west on a straight road. The wind is blowing from 248° (measured from the
positive x-axis) at 10 mph. Is this wind more like a headwind or a tail wind? What is the
headwind/tailwind speed? What is the crosswind speed?
(Acts more like a headwind, headwind speed 9.27 mph, crosswind speed 3.75 mph)
12. A motorboat is traveling across a wide river headed 𝑁40°𝑊 at a constant speed of 10 mph. The river
current is 7 mph due west. How far west does the motorboat travel in 3 minutes? How far north does
it travel in 3 minutes?
(It travels 0.671 miles west and 0.383 miles north)
Distance and direction:
1. A steamer sails 100 miles east and the 40 miles on a heading of 𝑁60°𝐸. How far is the steamer from its
starting point, and at what angle has it traveled from its starting point?
(Distance 136.12 miles, 8.45° − 𝑓𝑟𝑜𝑚 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑑𝑢𝑒 𝑒𝑎𝑠𝑡)
2. A laser beam is aimed 15.95° above the horizontal at a mirror 11,648 meters away. It glances off the
mirror and continues for an additional 8570 meters at 11.44° above the horizon until it hits its target.
What is the resultant displacement of the beam to the target?
(Displacement 20,202.7 meters, angle of elevation, 𝜃 = 14.04°
3. A person walks 30 km west, then 20 km north, then at 15 km east. What is her displacement?
(25 km, 𝑁36.86°𝑊)
4. A hiker walks away from base camp 30 km north, then 20 km east, then 50 km south, then 40 km west,
then 45 km north, then 35 km east. Exhausted, he radios to base camp to come get him. Which way
and how far do his rescuers have to go to get him?
(29.15 km, 𝑁30.96°𝐸)
Force
1. A 30-newton force and a 35-newton force act on an object with an angle of 80° between them. Find
the magnitude of the resultant force and the angle made with the 35-newton force.
(Force is 49.90-newtons Angle is 36.31°)
2. Two forces act on an object with an angle of 29° between the two forces. One force is 87 lbs. and the
other force is 121 lbs. Find the magnitude of the resultant and the angle made with the 87 lbs. force.
(Force 201.55 lbs. Angle is 16.92°)
3. A sled on an inclined plane weighs 500 lb., and the plane makes an angle of 50° with the horizontal.
What force, perpendicular to the plane, is exerted on the plane by the sled?
(Force 321.39 lb.)
4. A weight of 15 lb. is placed on a smooth plane inclined at tan able of 24° with the horizontal. What
force pushing long the plane will just prevent the weight from slipping?
(Force 6.10 lb.)
5. What would be the largest weight a person could drag up a slope inclined 35° from the horizontal if
that person is able to pull with a force of 125 lb.?
(Force 217.93 lb.)
6. If a force of 25 lb. is required to move an 80-lb. sled up a hill, what angle does the hill make with the
horizontal?
(18.21°)
7. Three forces act on a point: 3 N at 0°, 4 N at 90°, and 5 N at 217°. What is the net force?
(1.40 N at 134.93° from the positive x-axis)
8. Two dogs are pulling on the same bone. One pulls to the west with 50 N of force. The other pulls to the
south with 30 N of force. What is the resultant force on the bone?
(58.3 N, 𝑆59.04°𝑊 )
Work:
1. Amy uses 20N of force to push a lawn mower 10 meters. How much work does she do?
(Work 200 N-m)
2. How much work is done in lifting a 45-lb. child 8 feet off the ground? How much work is done if moving
an object 8 feet up if a 100-lb. force is applied in the direction of 2, 5 ?
(Work is 360 ft-lbs, work is 742.78 ft-lbs)