1. Vectors
Chapter 4

2. Flying Airplanes Requires an Understanding of Vectors
Cross Wind of 60 km/hr
Memphis
Plane traveling at 80 km/hr
Houston
Flying Airplanes Requires
an Understanding of Vectors

3. Steering a Boat Requires an Understanding of Vectors
Campsite
Across the Shore
Boat Dock
You can’t just steer your boat
directly towards the campsite
when the current is flowing east

4. Vectors
A vector has both magnitude and direction. (20 m East, 40 m/s North)
Vectors are represented by arrows.
An arrow will show the vector’s:
magnitude: Length of the vector
direction: Angle of the vector measured from east.
A resultant vector:
One vector that acts as the sum of two or more vectors.
Also called the Displacement Vector

5. Vector Direction
Direction of a vector is measured by its angle from East.
East is 0 degrees
North is 90 degrees
West is 180 degrees
South is 270 degrees
Sometimes direction is given as so many degrees _______ of ________. ( North of East, etc.)
Substitute “from” for “of”.

6. Vectors://

7. 2 Methods of Vector Addition
Graphical method – align vectors tip to tail and then draw the resultant from the tail of the first to the tip of the last.
Analytical- Algebraic method
Both give displacement or final position from the point of origin.

8. Head and Tail Winds: Tip to Tail
Head Wind
20 km/hr
100 km/hr
120 km/hr
80 km/hr
100 km/hr
20 km/hr
A tail wind increases velocity and a head wind decreases velocity

9. Pythagorean Theorem

10. A man walks 11 km N, then turns and walks 11 km E
A man walks 11 km N, then turns and walks 11 km E. What is his final displacement from his starting point?
The final displacement is found by solving for the length of R using the pythagorean theorem.
This only works for RIGHT TRIANGLES- must have a 90 degree angle!!

11. Algebraic Method using
Trig Functions
Sine θ = opposite/hypotenuse
Cosine θ = adjacent/hypotenuse
Tangent θ = opposite/adjacent θ

12. Trigonometric Functions
Sin θ = opposite/hypotenuse Soh
Cos θ = adjacent/hypotenuse Cah
Tan θ = opposite/adjacent Toa
a2 + b2 = c2
sin-1, cos-1, and tan-1
functions give θ
With any 2 values, you can
find all sides and all angles θ

13. Keys to Successful Graphical Addition
Clearly define your scale factor
Always add tip to tail

14. Sample Problems
Using a treasure map, a pirate walks 35 m east then 15 m north. What single line could the pirate have taken instead?
A pilot sets a plane’s controls at 250 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity?

15. Sample Problems
Using a treasure map, a pirate walks 35 m east then 15 m north. What single line could the pirate have taken instead?
A2 + b2 = c2
C= 38.1 m

16. A pilot sets a plane’s controls at 250 km/hr to the north
A pilot sets a plane’s controls at 250 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity?
a2 + b2 = c2

17. Writing Vectors in Polar Coordinates
A polar vector has both a magnitude [R] and a direction [degrees from east]
A Polar vector is written in polar coordinates [R,θ]
Example: [50 m/s, 40°]
Polar vectors must be changed to rectangular form (x,y) in order to be added.
50 m/s
θ = 40°

18. Changing from Polar to Rectangular
Since we always measure our angles from east (0 degrees) when in polar format, we can use the following formulas:
x = R cos θ
y = R sin θ

19. Example Problems
Convert the following polar coordinates to rectangular coordinates:
54 m/s, θ = 60 degrees
4.5 N, θ = 235 degrees
x = R cos θ
y = R sin θ

20. Changing from Rectangular to Polar
A Rectangular coordinate shows the x and y components of a vector.
Example: [38.3 m/s, 32.1 m/s]
For converting from rectangular to polar, we use the following formulas:
x2 + y2 = R2
θ = tan-1 (y/x)

21. Example Problems
Convert the following rectangular coordinates to polar coordinates:
(36 m/s, 22 m/s)
(-60 m, 35 m)
x2 + y2 = R2
θ = tan-1 (y/x)

22. Warm Ups
A ladder is used to climb a building that is 9 meters tall. The ladder makes an angle of 65 degrees with the ground. How long is the ladder?
A skateboard ramp is 3.3 meters long and makes an angle of 35 degrees with the ground. How high is the skateboarder when she leaves the ramp?
x = R cos θ
y = R sin θ

23. Adding Polar Vectors
Convert each polar vector to rectangular coordinates x = R cos θ y = R sin θ
Add all the x coordinates to get a single x coordinate. Add all the y coordinates to get a single y coordinate.
Convert back to polar coordinates.
x2 + y2 = R θ = tan-1 (y/x)

24. Warm Ups Convert to rectangular coordinates [15, 120°]
Convert to polar coordinates (4.2,-6.8)
Add [30, 115°] and [18, 255°]
An airplane flying due south at 230 km/hr experiences a crosswind from the east of 35 km/hr. What is the magnitude and direction of the resultant velocity?

25. Quick Review
How do you know if a pair of coordinates is polar or rectangular?
Which formulas do you use to convert from polar to rectangular?
Which formulas do you use to convert from rectangular to polar?
What are the three steps for adding polar vectors?

26. Warm Ups
Add the following vectors graphically to find the magnitude and direction. Then, add them analytically to find magnitude and direction.
3 km to the East. 6 km at 220° from East. 4 km at 150° from East.
A mover is loading a refrigerator into a truck with a ramp that is 8 feet long at an angle of 25 degrees with the ground. How far off the ground is the back of the truck? How far away from the truck is the ramp positioned?

27. Trigonometric Functions
Angle 1
Sin θ = 6/10 = .60
Cos θ = 8/10 = .80
Tan θ = 6/8 = 0.75
Angle 2
Sin θ = 8/10 = .80
Cos θ = 6/10 = .60
Tan θ = 8/6 = 1.33
Use sin-1 to get θ
Sine θ = opposite/hypotenuse
Cosine θ = adjacent/hypotenuse
Tangent θ = opposite/adjacent

28. 2 Methods of Vector Addition
2. Analytical Method – use Pythagorean Theorem and SOH CAH TOA
Sin = Opp / Hyp
Cos = Adj / Hyp
Tan = Opp / Adj