1. Chapter 4 Vector Addition
When handwritten, use an arrow:
When printed, will be in bold print: A
When dealing with just the magnitude of a vector in print, an italic letter will be used: A

2. Chapter 4 Vector Addition
Equality of Two Vectors
Two vectors are equal if they have the same magnitude and the same direction
Movement of vectors in a diagram
Any vector can be moved parallel to itself without being affected

3. Chapter 4 Vector Addition
Negative Vectors
Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions)
A = -B
Resultant Vector
The resultant vector is the sum of a given set of vectors

4. Chapter 4 Vector Addition
When adding vectors, their directions must be taken into account
Units must be the same
Graphical Methods
Use scale drawings
Algebraic Methods
More convenient

5. Chapter 4 Vector Addition
The resultant is the sum of two or more vectors. Vectors can
be added by moving the tail of one vector to the head of another vector without changing the magnitude or direction of the vector.
Vector Addition
Note: The red vector R has the same magnitude and direction.

6. Chapter 4 Vector Addition
Multiplying a vector by a scalar number changes its length
but not its direction unless the scalar is negative. V 2V -V

7. Chapter 4 Vector Addition
If two vectors are added at right angles, the magnitude
can be found by using the Pythagorean Theorem
R2 = A2 + B 2 and the angle by
If two vectors are added at any other angle, the magnitude
can be found by the Law of Cosines
and the angle by the Law of Sines

8. Chapter 4 Vector Addition
8 meters
62+82=102
36°
6 meters
10 meters
The distance traveled is 14 meters and the displacement
is 10 meters at 36º south of east.

9. Chapter 4 Vector Addition
A hiker walks 3 km due east, then makes a 30° turn north of east
walks another 5 km. What is the distance and displacement of the
hiker?
The distance traveled is 3 km + 5 km = 8 km
R2 = *3*5*Cos 150°
R2 = =60
R = 7.7 km R
5 km
3 km
The displacement is ° north of east

10. Chapter 4 Vector Addition
Add the following vectors and determine the resultant.
3.0 m/s, 45 and 5.0 m/s, 135
5.83 m/s, 104

11. Chapter 4 Vector Addition

12. Chapter 4 Vector Addition
A boat travels at 30 m/s due east across a river that is 120 m wide
and the current is 12 m/s south. What is the velocity of the boat
relative to shore? How long does it take the boat to cross the river?
How far downstream will the boat land?
30 m/s
30 m/s
12 m/s
12 m/s
= ° downstream.
The speed will be
The time to cross the river will be t = d/v = 120 m / 30 m/s = 4 s
The boat will be d = vt = 12 m/s * 4 s = 48 m downstream.

13. Chapter 4 Vector Addition
Examples

14. Chapter 4 Vector Addition

15. Chapter 4 Vector Addition

16. Chapter 4 Vector Addition
Add the following vectors and determine the resultant.
6.0 m/s, 225 m/s, 90
4.80 m/s, 

17. Chapter 4 Vector Addition
Add the following vectors and determine the resultant.
6.0 m/s, 225 m/s, 90
R2 = – 2*2*6*cos 45
R2 = –24 cos 45
R2 = 40 – = 23
R = 4.8 m R
6 m
2 m
45°
R = 

18. Chapter 4 Vector Addition
A component is a part
It is useful to use rectangular components
These are the projections of the vector along the x- and y-axes

19. Chapter 4 Vector Addition
The x-component of a vector is the projection along the x-axis
The y-component of a vector is the projection along the y-axis
Then,

20. Chapter 4 Vector Addition
The previous equations are valid only if θ is measured with respect to the x-axis
The components can be positive or negative and will have the same units as the original vector
The components are the legs of the right triangle whose hypotenuse is A
May still have to find θ with respect to the positive x-axis

21. Chapter 4 Vector Addition
Choose a coordinate system and sketch the vectors
Find the x- and y-components of all the vectors
Add all the x-components
This gives Rx:

22. Chapter 4 Vector Addition
Add all the y-components
This gives Ry:
Use the Pythagorean Theorem to find the magnitude of the Resultant:
Use the inverse tangent function to find the direction of R:

23. Chapter 4 Vector Addition
Vector components is taking a vector
and finding the corresponding
horizontal and vertical components.
Vector resolution A Ay Ax

24. Chapter 4 Vector Addition
A plane travels 500 km at 60°south of east.
Find the east and south components of its
displacement. de
de= 500 km *cos 60°= 250 km
60° ds
ds= 500 km *sin 60°= 433 km
500 km