1. Vector Addition and Subtraction
Graphic Methods

2. Vectors
Quantities having both magnitude and direction are vector quantities.
Vectors can be represented by an arrow- tipped line segment.
Examples of vectors:
Velocity
Acceleration
Displacement
Force

3. 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).
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.

4. Using the Graphic 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.
Any number of vectors can be added using this method.

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

6. Vector Subtraction
The process of vector subtraction is the same as vector addition.
Addition:
Subtraction: where -B has the same magnitude as B but its direction is reversed by 180o. 6

7. 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?

8. 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

9. 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°

10. If the river is 80.0 m wide, how long will the boat take to cross the river?
v = Δd / Δt
Δt = Δd / v
80.0 m
= 10.0 s
8.00 m/s

11. How far down stream will the boat be carried in this amount of time?
v = Δd / Δt
Δd = vΔt
5.00 m/s (10.0 s)
= 50.0 m

12. 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

13. 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°

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

15. 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

16. 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

17. 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

18. 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.

19. 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.

20. 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.

21. 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.

22. Vector Resolution
y = h sin Θ
x = h cos Θ h y Θ x
+ ++
- +-

23. Components of Force: y x

24. 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
θ = arctan 6.74/0.09
= 89.2°

25. Advantages of the Component Method:
Can be used for any number of vectors.
All vectors are added at one time.
Only a limited number of mathematical equations must be used.
Least time consuming method for multiple vectors.

26. Solve the following problem using the component method.
10 km at 30°
6 km at 120°