1. Preview Section 1 Introduction to Vectors Section 2 Vector Operations
Section 3 Projectile Motion
Section 4 Relative Motion

2. Vector Operations
Use a traditional x-y coordinate system as shown below on the right.
The Pythagorean theorem and tangent function can be used to add vectors.
More accurate and less time-consuming than the graphical method
Direction means north, south, east, west, up, or down. It does not mean increasing or decreasing. So even though the temperature may be going “up,” it is really just increasing and has no direction.

3. Pythagorean Theorem and Tangent Function
Remind students that the Pythagorean theorem can only be used with right triangles.

4. Vector Addition - Sample Problems
12 km east + 9 km east = ?
Resultant: 21 km east
12 km east + 9 km west = ?
Resultant: 3 km east
12 km east + 9 km south = ?
Resultant: 15 km at 37° south of east
12 km east + 8 km north = ?
Resultant: 14 km at 34° north of east
For the first two items, have students predict the answer before showing it. They generally have no trouble with these two problems. Point out that the process is the same if it is km/h or m/s2. Only the units change. These problems do not require trigonometry because the vectors are in the same direction (or opposite directions).
For the third problem, most students will probably remember the Pythagorean theorem and get the magnitude, but many will fail to get the direction or will just write southeast.
Show students how to use the trig identities to determine the angle. Then, explain why it is south of east and not east of south by showing what each direction would look like on an x-y axis. If they draw the 9 km south first and then add the 12 km east, they will get an answer of 53° east of south (which is the same direction as 37° south of east). After your demonstration, have students solve the fourth problem on their own, and then check their answers. Review the solution to this problem also.
Insist that students place arrows on every vector drawn. When they just draw lines, they often draw the resultant in the wrong direction.
You might find the PHet web site helpful ( If you go to the Math simulations, you will find a Vector Addition (flash version). You can download these simulations so your access to the internet is not an issue. You can show both the resultant and components using this simulation. Students could also use this at home to check their solutions to problems.

5. Resolving Vectors Into Components
Review these trigonometry definitions with students to prepare for the next slide (resolving vectors into components).

6. Resolving Vectors into Components
Opposite of vector addition
Vectors are resolved into x and y components
For the vector shown at right, find the vector components vx (velocity in the x direction) and vy (velocity in the y direction). Assume that that the angle is 20.0˚.
Answers:
vx = 89 km/h
vy = 32 km/h
Review the first solution with students, and then let them solve for the second component.

7. Adding Non-Perpendicular Vectors
Four steps
Resolve each vector into x and y components
Add the x components (xtotal = x1 + x2)
Add the y components (ytotal = y1 + y2)
Combine the x and y totals as perpendicular vectors
Explain the four steps using the diagram.
Show students that d1 can be resolved into x1 and y1 . Similarly for d2. Then, the resultant of d1 and d2 (dashed line labeled d) is the same as the resultant of the 4 components.

8. Adding Vectors Algebraically
Click below to watch the Visual Concept.
Visual Concept

9. Classroom Practice Displacement
Following the directions of the path, a pirate walks 45.0 m north, then turns and walks 7.5 m east. What single straight line displacement could the pirate have taken to reach the destination.
Answer
4.5m at 9.5° E of N
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

10. Classroom Practice Displacement
Emily passes a soccer ball 6.0 m directly across the field to Kara, who then kicks the ball 14.5m directly down the field to Luisa. What is the ball’s total displacement as it travels between Emily and Luisa?
Answer
15.7m at 22° to the side of down the field
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

11. Classroom Practice Displacement
A humming bird flies 1.2 m along a straight path at a height of 3.4 m above the ground. Upon spotting a flower below, the humming bird drops directly downward 1.4 m to hover in front of the flower. What is the hummingbird’s displacement?
Answer
1.8 m at 49° below the horizontal
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

12. Classroom Practice Resolving Vectors
Find the horizontal and vertical component of the 125 m displacement of a superhero who flies down from the top of a tall building at an angle of 25 ° below the horizontal.
Answer
110 m
-53 m
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

13. Classroom Practice Resolving Vectors
Find the horizontal and vertical component of the child’s toboggan ride down a hill if the angle of the hill is 30.5 ° to the horizontal and the hill is 23m long.
Answer
19.8 m
-11.7 m
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

14. Classroom Practice Resolving Vectors
Find the horizontal and vertical component of the truck’s velocity, if the truck drives up a hill with a 15° incline at a constant speed of 22 m/s.
Answer
21m/s
5.7m/s
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

15. Classroom Practice Algebraically
A football player runs directly down the field for 35 m before turning to the right at an angle of 25° from his original direction an running an additional 15 m before getting tackled. What is the total displacement of the player
Answer
49 m at 7.3° to the right of down field
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

16. Classroom Practice Algebraically
A plane travels 25 km at an angle of 35 to the ground, then changes direction and travels 515 km at an angle of 22° to the ground. What is the displacement of the plane?
Answer
540,000 m at 22° above the horizontal
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

17. Classroom Practice Algebraically
During a rodeo a clown runs 8.0 m north, turns 35 ° east of north, and runs 3.5 m. then , after waiting for the bull to come near, the clown turns due east and runs 5.0 m to exit the arena. What is the clown’s displacement?
Answer
13.5 m at 37° north of east
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

18. Classroom Practice Resolving Vectors
Find the magnitude resultant velocity.
A fish swimming a t 3.0 m/s across a river that moves at 5.0 m/s?
What is the direction?
Answer
5.8 m/s
59° downriver from its intended path
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

19. Classroom Practice Resolving Vectors
Find the magnitude resultant velocity.
A surfer travelling at 1.0 m/s across a wave that is traveling at 6.0 m/s?
What is the direction?
Answer
6.1 m/s
9.5° from the direction the wave is traveling
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

20. Classroom Practice Resolving Vectors
Find the component vectors along the directions noted in parentheses.
A car displaced northeast by 10.0 km (north and east
Answer
7070 m, 7070 m
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

21. Classroom Practice Resolving Vectors
Find the component vectors along the directions noted in parentheses.
A duck accelerating away from a hunter at 2.0 m/s2 at an angle of 35° to the ground (horizontal and vertical)
Answer
1.6 m/s2, 1.1 m/s2
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

22. Classroom Practice Resolving Vectors
Find the component vectors along the directions noted in parentheses.
A submarine moving at 10.0 m/s toward the surface at an angle of 35 ° to the ground (horizontal and vertical)
Answer
8.2 m/s, 5.7 m/s
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.

23. Classroom Practice Resolving Vectors
Find the resultant displacement of a fox searching for mice in a prairie. First the fox heads 55° north of west for 10.0 m then it turns and heads west for 5.0m.
Answer
13.5 m at 37° north of east
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.