1. Vectors and Vector Operations
“What’s our vector, Victor?”

2. Introduction to Vectors
Scalar - a quantity that has magnitude but no direction
Examples: volume, mass, temperature, speed
Vector - a quantity that has both magnitude and direction
Examples: acceleration, velocity, displacement, force
Emphasize that direction means north, south, east, west, up, or down. It does not mean increasing or decreasing. Even though the temperature may be going “up”, it is really increasing and has no direction. To further emphasize the distinction, point out that it is meaningless to talk about the direction of temperature at a particular point in time, while measurements such as velocity have direction at each moment.

3. Vectors are generally drawn as arrows.
Vector Properties
Vectors are generally drawn as arrows.
Length represents the magnitude
Arrow head shows the direction
The resultant is the sum of two or more vectors

4. Finding the Resultant Graphically
Method
Draw each vector in the proper direction.
Establish a scale (i.e. 1 cm = 2 m) and draw the vector the appropriate length.
Draw the resultant from the tip of the first vector to the tail of the last vector.
Measure the resultant.
The resultant for the addition of a + b is shown to the left as c.
Ask students if a and b have the same magnitude. How can they tell?

5. Tail to Head Head to Tail Vector Addition
Stress that the order in which they are drawn is not important because the resultant will be the same.

6. Vector Addition
Vectors can be moved parallel to themselves without changing the resultant.
the red arrow represents the resultant of the two vectors
Stress that the order in which they are drawn is not important because the resultant will be the same.

7. Vectors can be added in any order.
Vector Addition
Vectors can be added in any order.
The resultant (d) is the same in each case
Subtraction is simply the addition of the opposite vector.

8. Tail to Tail Head to Head Vector Subtraction
Stress that the order in which they are drawn is not important because the resultant will be the same.

9. Vector Subtraction (Alternate)
Tail to Head
Tail to Head
Stress that the order in which they are drawn is not important because the resultant will be the same.

10. Sample Resultant Calculation
A toy car moves with a velocity of +.80 m/s across a moving walkway that travels at +1.5 m/s. Use graphing methods to find the resultant speed of the car.
Use this to demonstrate the graphical method of adding vectors. Use a ruler to measure the two components and determine the scale. Then determine the size and direction of the resultant using the ruler and protractor. This would make a good practice problem for Section 2, when students learn how to add vectors using the Pythagorean theorem and trigonometry.

11. Finding the Resultant Graphically
Method
Draw each vector in the proper direction.
Establish a scale (i.e. 1 cm = 2 m) and draw the vector the appropriate length.
Draw the resultant from the tip of the first vector to the tail of the last vector.
Measure the resultant.
The resultant for the addition of a + b is shown to the left as c.
Ask students if a and b have the same magnitude. How can they tell?

12. Graphing Vectors Practice
A bird flies at 16.5 m/s at an angle of 40o north of east, then flies at 18.2 m/s at an angle of 200 south of east. Use graphical methods to determine the magnitude and direction of the bird’s velocity.

13. Chapter 3
Section 2

14. 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.

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

16. Pythagorean Theorem for Vectors
Hypotenuse
Opposite
Adjacent

17. 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.

18. Group Practice
An airplane flies due east at 325 km/h. The wind is blowing at 75 km/h north. What is the airplane’s actual direction and speed?

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

20. Resolving Vectors into Components
Opposite of vector addition
Breaks a vector down into two smaller vectors that make a right triangle
Makes the original vector a resultant

21. Resolving Vectors into Componentsx y 

22. Let’s consider 2 vectors: 23o east of north and 35o north of east
Let’s consider 2 vectors: 23o east of north and 35o north of east. Here is the process (steps) to break them into components:
1. Sketch the vectors. Include a lightly-drawn or dotted-line coordinate (x- and y-axis) system: Identify the trig function (sine or cosine) you need for the x- component: a) If angle “opens” to left or right, the x-component is the adjacent side Use the cosine function:
b) If the angle “opens” up or down, the x-component is the opposite side Use the sine function.

23. 3. Identify the trig function (sine or cosine) you need for the y- component: a) If angle “opens” to left or right, the y-component is the opposite side Use the sine function:
b) If the angle “opens” up or down, the y-component is the adjacent side Use the cosine function.
Note: If the angle “opens” into quadrants II, III, or IV, one or both of the components will have a negative sign!

24. Group Practice
Find the components of the velocity of a helicopter traveling 95 km/hr at an angle of 35o to the ground.

25. More practice
A truck carrying a film crew must be driven at the correct velocity to enable the crew to film the underside of a biplane. The plane flies at 95 km/hr at an angle of 20o relative to the ground. Calculate the velocity the truck has to be driven in order to stay with the plane.

26. 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.

27. Adding Non-Perpendicular Vectors
y2 d
y1
x1
x2
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.

28. d2 y2 d d1 y1 x1 x2 d1 d1cos1 d1sin1 d2 d2cos2 d2sin2 d
Vector
x-component
y-component d1
d1cos1
d1sin1 d2
d2cos2
d2sin2 d
 x-comp
 y-comp

29. Group Practice
A camper walks 4.5 km at 45° north of east and then walks 4.5 km due south. Find the camper’s total displacement.
Answer
3.5 km at 22° S of E
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.

30. 4.5 km 4.5 km 45o 1 2 R (4.5 cos) km 3.2 km (4.5 sin) km 3.2 km
vector
x-component
y-component 1 2 R
(4.5 cos) km
3.2 km
(4.5 sin) km
3.2 km
4.5 km
3.2 km
1.3 km

31. 4.5 km
4.5 km
45o
vector
x-component
y-component R
3.2 km
1.3 km

32. Group Practice
A hiker walks 27.0 km from her base camp at 35o south of east. The next day, she walks 41.0 km in a direction 65o north of east and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.
Answer: 45 km and 29o north of east

33. Practice
A plane flies 118 km at 15o south of east and then flies 118 km at 35o west of north. Find the magnitude and direction of the total displacement of the plane.
81 km at 55o north of east