Vectors obey vector algebra rather than ordinary algebra. Vectors have both magnitude and direction, while scalars only have magnitude. Common vectors include displacement, velocity, acceleration, force, and momentum. Vectors can be represented using arrows, with the length representing magnitude and the arrowhead representing direction. Vector addition involves drawing the vectors tip-to-tail to find the resultant vector, rather than simply adding their magnitudes. Vectors can also be added by adding their corresponding components.

1. VECTORSslides courtesy of Dr. Gammag

2. Vectors obey vector algebra, not ordinary algebra.

3. 1. How many chairs are in this room?
4. How do you get to your house from here?
2. How much does the textbook cost?
3. How long does it take to study for the finals?
5. What is the wind velocity?
6. What is your blood pressure?
Which of the following are vectors?
7. What is your car’s mileage?

4. Scalars – have magnitude only
Vectors – have magnitude and direction

5. Scalars – obey (ordinary) algebra
Vectors – obey vector algebra

6. Vector Scalar
Displacement
Velocity
Acceleration
Force
Momentum
Torque
Electric Field
Distance
Speed
Temperature
Mass
Energy
Time
Work

7. Arrows can represent vectors
length: magnitude
arrowhead : direction

8. Vector Example: Displacement

9. Equal Vectors Negative Vectors

10. Direction
Compass 72º North of East = 72º NE
= 72º N from E = 18º E of N
Clock 5:00 = −60o
Cartesian plane
45o, −45o,
+ direction: counterclockwise

11. N
EW
S
375 m, 30° S of W
343 m, 26° W of N
531 m, 82° E of S
437 m, 70° N of E

12. Multiplying vector with a scalar
A vector quantity 𝑨 can be multiplied by a scalar quantity 𝑐:
𝑩 = 𝑐𝑨
where 𝑩 is another vector.

13. Vector Addition : Drawing Method
AVOID COMMON MISTAKE:
The sum of two vectors is
not the sum of their magnitudes!

14. =
Vector addition commutes.

15. Vector subtraction = addition of negative vector

16. Vector sum of two vectors at right angles
A farmer is checking if his field is ready for the
next planting season. He walks 1.00 km north
and then 2.00 km west. How far and in what
direction is he from his starting point?
He travelled 2.24 km, 64.4o West of N

17. COMPONENTS
y
x
𝑨
𝐴 𝑦
𝐴 𝑥
𝜃
𝜙 𝐴 𝑦
𝐴 𝑥
The components are the shadows or projections of a
vector at specific directions.
The components can be expressed in terms of an
angle or its complement. Recall SOH CAH TOA.

18. COMPONENTS can be negative.

19. COMPONENTS can be negative.

20. Finding components

21. angle 𝜽 𝑥-component 𝑦-component
measured from 𝑥-axis 𝐷 𝑥 = 𝐷 cos 𝜃 𝐷 𝑦 = 𝐷 sin 𝜃
measured elsewhere SOH CAH SOH CAH

22. UNIT VECTORS  for pointing direction
 obey the right-hand rule

23. UNIT VECTORS

24. 𝑨 = 𝐴 𝑥 𝒊 + 𝐴 𝑦 𝒋
𝑩 = 𝐵𝑥 𝒊 + 𝐵𝑦 𝒋
ADDITION BY COMPONENTS (2D)
𝑹 = 𝑨 + 𝑩
𝑹 = 𝐴 𝑥 𝒊 + 𝐴 𝑦 𝒋 + 𝐵𝑥 𝒊 + 𝐵𝑦 𝒋
𝑹 = 𝐴 𝑥 + 𝐵𝑥 𝒊 + 𝐴 𝑦 + 𝐵𝑦 𝒋
The sum of vectors is also
called the resultant or the
net vector.
𝑹 = 𝑅 𝑥 𝒊 + 𝑅 𝑦 𝒋
Add components of
the same direction.

25. ADDITION BY COMPONENTS (3D)
Add components of
the same direction.

26. ADDITION BY COMPONENTS
12.7 m, 39o
west of north