1. PHY 1151 Principles of Physics I
Chapter 3
Vectors in Physics
PHY 1151 Principles of Physics I

2. PHY 1151 Principles of Physics I
Outline
Vectors and Scalar Quantities
Graphical Representation of Vectors
Some properties of Vectors
Equality of Two Vectors
Adding Vectors (Graphically)
Subtracting Vectors (Graphically)
Multiplying a Vector by a Scalar
PHY 1151 Principles of Physics I

3. PHY 1151 Principles of Physics I
Scalar Quantities
Scalar quantity: A scalar quantity is completely specified by a single value with appropriate units and has no direction.
Examples: temperature, volume, mass, time intervals, speed
PHY 1151 Principles of Physics I

4. PHY 1151 Principles of Physics I
Vector Quantities
Vector quantity: A vector quantity is completely specified by a number with appropriate units (magnitude) plus a direction.
A vector quantity has both magnitude and direction.
Examples: position, displacement, velocity, acceleration, force
We use boldface letter, such as A, to represent a vector quantity. The magnitude of the vector A is written either as A or |A|.
PHY 1151 Principles of Physics I

5. Graphical Representation of Vectors
Example: a force vector of magnitude = 3 N and directed to the right can be represented as follows
Vectors can be represented graphically using arrows
The direction of the arrowhead represents the direction of the vector.
The length of the arrow represents the magnitude of the vector. F
Scale: 1 cm 1N
PHY 1151 Principles of Physics I

6. PHY 1151 Principles of Physics I
Examples of Vectors x y x y r
Position Vector r D
Displacement Vector D as a particle moves from A to B
PHY 1151 Principles of Physics I

7. Equality of Two Vectors
Two vectors A and B are defined to be equal if they have the same magnitude and point in the same direction.
A = B only if A = B and A and B point in the same direction.
All the vectors in this figure are equal even though they have different starting points.
PHY 1151 Principles of Physics I

8. PHY 1151 Principles of Physics I
Adding Vectors
Graphical representation of vector addition
The resultant vector R = A + B is the vector drawn from the tail of A to the tip of B
The resultant vector R = A + B + C + D is the vector drawn from the tail of the first vector to the tip of the last vector
PHY 1151 Principles of Physics I

9. Example: Adding Vectors
If you walked 3.0 m toward the east and then 4.0 m toward the north. Find the magnitude and direction of your total displacement.
Two ways to solve the problem:
Graphical method: Using graph paper and a protractor.
Algebra method: Calculation using algebra
PHY 1151 Principles of Physics I

10. PHY 1151 Principles of Physics I
Subtracting Vectors
Negative of a vector: The vectors A and –A have the same magnitude but point in opposite directions. Thus, A + (-A) = 0.
Subtracting vectors:
We define operation A – B as vector – B added to vector A: A – B = A + (- B)
PHY 1151 Principles of Physics I

11. Multiplying a Vector by a Scalar
If vector A is multiplied by a positive scalar quantity m, then the product mA is a vector, which has the same direction as A and magnitude mA.
If vector A is multiplied by a negative scalar quantity -m, then the product -mA is directed opposite A and has magnitude mA. F 2F
- 2F
PHY 1151 Principles of Physics I

12. PHY 1151 Principles of Physics I
Quick Quiz
If vector B is added to vector A, under what condition does the resultant vector A + B have magnitude A + B? Under what conditions is the resultant vector equal to zero?
PHY 1151 Principles of Physics I

13. PHY 1151 Principles of Physics I
Homework
See online homework assignment at
PHY 1151 Principles of Physics I