1. Graphical Analytical Component Method
Vector Addition

2. Math sTUFF
What does an ordered pair mean in math? Ex:(2,3)

3. Vectors Quantities having both magnitude and direction
Magnitude: How much (think of it as the length of the line)
Direction: Which way is it pointing?
Can be represented by an arrow-tipped line segment
Examples:
Velocity
Acceleration
Displacement
Force

4. Question:
Compare the two vectors. What makes them different

5. Answer
Direction
The magnitude or length is exactly the same

6. Vector Terminology
Two or more vectors acting on the same point are said to be concurrent vectors.
The sum of 2 or more vectors is called the resultant (R).
A single vector that can replace concurrent vectors
Any vector can be described as having both x and y components in a coordinate system.
The process of breaking a single vector into its x and y components is called vector resolution.

7. More Vector Terminology
Vectors are said to be in equilibrium if their sum is equal to zero.
A single vector that can be added to others to produce equilibrium is call the equilibrant (E).
Equal to the resultant in magnitude but opposite in direction.
E + R = 0
E = - R
E = 5 N R = 5 N
at 180 ° at 0°

8. What is the resultant of the following vectors?
E= 10 N at 0 degrees
R = 20 N at 0 degrees

9. Answer
30 N at 0 degrees

10. Question 20 N at 45 degrees 10 N at 225 degrees
Do not get freaked out by the angles, Think about it for a second.

11. Answer
10 N at 45 degrees

12. Using the Graphical Method of Vector Addition:
Vectors are drawn to scale and the resultant is determined using a ruler and protractor.
Vectors are added by drawing the tail of the second vector at the head of the first (tip to tail method).
The order of addition does not matter.
The resultant is always drawn from the tail of the first to the head of the last vector.

13. Example Problem
A 50 N force at 0° acts concurrently with a 20 N force at 90°. R  
R and  are equal on each diagram.

14. Adding vectors!

15. Question: Add these two vectors togetherb

16. Answer b a
R= a+b

17. Motion Applications
Perpendicular vectors act independently of one another.
In problems requesting information about motion in a certain direction, choose the vector with the same direction.

18. Example Problem: Motion in 2 Dimensions
A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s.
What is the resultant velocity of the boat?

19. A boat heads east at 8. 00 m/s across a river flowing north at 5
A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s.
5.00 m/s N
8.00 m/s E
River width

20. What is the resultant velocity of the boat?
Draw to scale and measure.
5.00 m/s N
8.00 m/s E
R = 9.43 m/s at 32°

21. Advantages and Disadvantages of the Graphical Method
Can add any number of vectors at once
Uses simple tools
No mathematical equations needed
Must be correctly draw to scale and at appropriate angles
Subject to human error
Time consuming

22. Solving Vectors Using the Analytical Method
A rough sketch of the vectors is drawn.
The resultant is determined using:
Algebra
Trigonometry
Geometry

23. Quick Review Right Triangle c is the hypotenuse B c2 = a2 + b2 c
sin = o/h cos = a/h tan = o/a a
A + B + C = 180°
B = 180° – (A + 90°) C A b
tan A = a/b
tan B = b/a

24. These Laws Work for Any Triangle.
A + B + C = 180° C
Law of sines:
a = b = c
sin A sin B sin C b a B A c
Law of cosines:
c2 = a2 + b2 –2abCos C

25. Use the Law of: Sines when you know: Cosines when you know:
2 angles and an opposite side
2 sides and an opposite angle
Cosines when you know:
2 sides and the angle between them

26. For right triangles: Draw a tip to tail sketch first.
To determine the magnitude of the resultant
Use the Pythagorean theorem.
To determine the direction
Use the tangent function.

27. To add more than two vectors:
Find the resultant for the first two vectors.
Add the resultant to vector 3 and find the new resultant.
Repeat as necessary.

28. Advantages and Disadvantages of the Analytical Method
Does not require drawing to scale.
More precise answers are calculated.
Works for any type of triangle if appropriate laws are used.
Can only add 2 vectors at a time.
Must know many mathematical formulas.
Can be quite time consuming.

29. Solving Vector Problems using the Component Method
Each vector is replaced by 2 perpendicular vectors called components.
Add the x-components and the y-components to find the x- and y-components of the resultant.
Use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant.

30. Vector Resolution
y = h sin 
x = h cos  h y  x
+ ++
- +-

31. Components of Force: y x

32. Question What are the components of the following force
12 degrees North of West

33. Answer West is 180 degrees to 12 degrees north of west is 168 degrees
The X component is N
The Y component is 5.20N
You can confirm you answer –X and +Y would be found in the second quadrant on a graph so this answer makes sense

34. Example: x y + 0.09 + 6.74 6 N at 135° 5 N at 30°
5 cos 30° = +4.33
5 sin 30° = +2.5
6 cos 45 ° =
6 sin 45 ° =
+ 0.09
+ 6.74
5 N at 30°
R = (0.09)2 + (6.74)2
= 6.74 N
 = arctan 6.74/0.09
= 89.2°

35. Tangent Function
The tangent function has 2 places that it is not defined (you get an error on your calculator)
90 degrees and 270 degrees
The x and y components tell you the angle range
Angle Range
Operation
0 to 90
Nothing special needed
90 to 180
Add 180 degrees
180 to 270
270 to 360
Add 360 degrees

36. Question: Critical THinking
My X component was negative and my y component was negative as well. My calculator told me that my answer was 22 degrees. What is my true angle?

37. Answer
My evidence:
Negative X
Negative Y
We are in quadrant three (between 180 degrees and 270)
I got 22 degrees, so I must take to get 202 degree as my angle! Using my tangent rules

38. Solve the following problem using the component method.
10 km at 30
6 km at 120

39. Adding Vectors To find the magnitude: pythagorean theorum
X component
Y component
6 120 degrees
6 km * cos(120) =
6 km * sin(120) =
10 30 degrees
10 km * cos (30) =
10 km * sin(30) =
Add them together
To find the magnitude: pythagorean theorum
To find the direction:
1. Take into account if either X or y is + or –
2. Use any trig function SOH CAH TOA to find angle

40. Critical Thinking Question 2
I get a positive x and a negative y component when I add them together.
What degree range is my angle in?

41. Answer X is positive so that can only mean either quadrant 1 or 4
Y is negative so that means you have to get quadrant 4 as your answer
270 to 360 degrees

42. Notes
Make sure that all angles are measured from the x axis (0 degrees)
Report both the magnitude and the direction otherwise the vector is wrong!
Keep track of signs, They give you a clue to where the angle of the vector actually is.

43. Adding two vectors Find the resultant magnitude:
X component
Y component
degrees
degrees
Resultant X and Y
Find the resultant magnitude:
Find the resultant direction: