1. VECTOR ADDITION

2. Vectors
Vectors Quantities have magnitude and direction and can be represented with;
Arrows
Sign Conventions (1-Dimension)
Angles and Definite Directions (N, S, E, W)

3. Vectors Every Vector has two parts. ------------------ Tail Head
A Head and a Tail. 
Tail Head

4. Vectors
One dimensional Vectors can use a “+” or “–” sign to show direction.
+ 50 m/s a m b
- 9.8 m/s2 i N h

5. Picking an Angle for 2-Dimensional Vectors
Pick the angle from the tail of the Vector.
Find a Definite Direction w.r.t. the Vector (N, S, E, W).
Write your angle w.r.t. the Definite Direction.

6. Addition of Scalars
When adding scalars, direction does not matter so we add or subtract magnitudes.
50 kg + 23 kg = 73 kg
25 s – 13 s = 12 s

7. Addition of Vectors
Adding vectors is more complicated because the direction affects how the vectors can be combined.
A a B l A d B m
A + B = ? A + B = ?

8. Addition of 1-Dimensional Vectors
One Dimensional Vectors are either in the same or opposite directions.
If the Vectors are in the same direction we add their magnitudes.
If the Vectors are in opposite directions we subtract their magnitudes.

9. Addition of 2-Dimensional Vectors
Addition of 2-Dimensional Vectors requires other methods.
Graphical Method
Mathematical or Triangle Method
Component Method (Best Method)

10. Properties of Vectors
The addition of vectors yields a RESULTANT (R) R = A + B
Order of addition does not matter!
A + B = B + A = R
A+B+C = C+B+A = B+C+A = R

11. Properties of Vectors
Vectors that are Perpendicular to each other are INDEPENDENT of each other.
Ex. Boat traveling perpendicular to the current.
Ex. Projectile thrown in the air.

12. Graphical Addition of Vectors R = A + B
Adding vectors Graphically has certain advantages and disadvantages.
Not a mathematical method

13. Graphical Addition of Vectors R = A + B
Advantages
You can add as many vectors as you want
Importance:
Vector Diagram
Disadvantages
Need a Ruler and a Protractor
Not the most accurate method

14. Graphical Addition of Vectors R = A + B
Pick an appropriate scale to draw vectors
Draw the First Vector (A) to scale
Draw the Next Vector (B) from the Head of the Previous Vector (A)

15. Graphical Addition of Vectors R = A + B
Draw the Resultant (R) from the Tail of the First to the Head of the Last
Measure the Resultant (R), length and direction, and use your scale to give your answer

16. Vectors are Relative
Vectors that are measured depend on their reference point or frame of reference
Velocities measured are relative w.r.t. the Observer
1-Dimensional (Ex. Bus Ride)
2-Dimensional (Ex. River Crossing)

17. Subtraction of Vectors R = A – B
Subtraction is the Addition of a Negative Vector
A - B = A + (-B)
So what does –B mean?
-B is the vector that has the same magnitude of B but in the opposite direction!

18. Subtraction of Vectors R = A – B
Ex. A = 50 N a
B = 40 N h -B = 40 N ,
A + B = ? A – B = ?
Does (A – B) = (B – A) ???
No!!!
A – B = -B + A = -(B – A)

19. Vector Addition: Triangle Method
Advantages
Mathematical Method (More Accurate)
Importance:
Working with Right Triangles
Disadvantages
Can only add two vectors at a time

20. Vector Addition: Triangle Method
1. Use a Vector Diagram to make your triangle
(see Graphical Method)
2. See what kind of Triangle you have

21. Vector Addition: Triangle Method
3. If you have a Right Triangle use:
Pythagorean Theorem
c2 = a2 + b2
Tangent Function
tan q = opposite side = O
adjacent side A

22. Vector Addition: Triangle Method
If you do not have a Right Triangle use:
Law of Cosines
c2 = a2 + b2 – 2abCos(C)
Law of Sines
Sin(A) = Sin(B) = Sin(C)
a b c

23. Perpendicular Components of a Vector
Every Vector can be split into two perpendicular components
A = Ax + Ay
Ax = Horizontal component of A
Ay = Vertical component of A

24. Perpendicular Components of a Vector
Ax and Ay are one dimensional vectors
Ax - a “+” d “-”
Ay - h “+” i “-”

25. To Find Perpendicular Components of a Vector
Since the components(Ax, Ay) are perpendicular, they form a right triangle with. Vector A
Use the sine and cosine functions to find Ax and Ay

26. To Find Perpendicular Components of a Vector
sin q = Opposite = O
Hypotenuse H
cos q = Adjacent = A

27. Component Method Advantages to the Component Method
Do not need a ruler or a protractor (More accurate)
Can add multiple vectors
Every triangle is a right triangle!!

28. Component Method
Break up each vector into its perpendicular components (Ax, Ay, Bx, By, Cx, Cy, etc,)
Add up all the x-components to get the x-component of Resultant (Rx = Ax + Bx + Cx + … etc)

29. Component Method
Add up all the y-components to get the y-component of Resultant (Ry = Ay + By + Cy + … etc)
Recombine Rx and Ry to make the Resultant, R (R = Rx + Ry) (Use the Triangle Method for Right Triangles)
Every triangle is a Right Triangle