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

2. PHY 1151 Principles of Physics I
Outline
Components of a vector
How to find the components of a vector if knowing its magnitude and direction
How to find the magnitude and direction of a vector if knowing its components
Express a vector in terms of unit vectors
Adding vectors using the Components Method
PHY 1151 Principles of Physics I

3. Components Method for Adding Vectors
The graphical method of adding vectors is not recommended when high accuracy is required or in three-dimensional problems.
Components method (rectangular resolution): A method of adding vectors that uses the projections of vectors along coordinate axes.
PHY 1151 Principles of Physics I

4. PHY 1151 Principles of Physics I
Components of a Vector
Components of a vector: The projections of a vector along coordinate axes are called the components of the vector.
Vector A and its components Ax and Ay
The component Ax represents the projection of A along the x axis.
The component Ay represents the projection of A along the y axis.
PHY 1151 Principles of Physics I

5. Find the Components of a Vector Given its Magnitude and Direction
If vector A has magnitude A and direction , then its components are
Ax = A cos
Ay = A sin
Note: According to convention, angle  is measured counterclockwise from the +x axis.
PHY 1151 Principles of Physics I

6. Signs of the Components Ax and AyII
III IV
The signs of the components Ax and Ay depend on the angle , or in which quadrants vector A lies.
Component Ax is positive if vector Ax points in the +x direction.
Component Ax is negative if vector Ax points in the -x direction.
The same is true for component Ay.
PHY 1151 Principles of Physics I

7. PHY 1151 Principles of Physics I
Example: Find the Components of a Vector
Find Ax and Ay for the vector A with magnitude and direction given by
(1) A = 3.5 m and  = 60°.
(2) A = 3.5 m and  = 120°.
(3) A = 3.5 m and  = 240°.
(4) A = 3.5 m and  = 300°.
PHY 1151 Principles of Physics I

8. Find the Magnitude and Direction of A Given its Components Ax and Ay
The magnitude and direction of A are related to its components through the expressions:
A = (Ax2 + Ay2)1/2
 = tan-1(Ay/Ax)
Note: Pay attention to the signs of Ax and Ay to find the correct values for .
PHY 1151 Principles of Physics I

9. Example: Find the Magnitude and Direction of a Vector
Find magnitude B and direction  for the vector B with components
(1) Bx = 75.5 m and By = 6.20 m.
(2) Bx = m and By = 6.20 m.
(3) Bx = m and By = m.
(4) Bx = m and By = m.
PHY 1151 Principles of Physics I

10. Express Vectors Using Unit Vectors
Unit vectors: A unit vector is a dimensionless vector having a magnitude of exactly 1.
Unit vectors are used to specify a given direction and have no other physical significance.
Symbols i, j, and k represent unit vectors pointing in the +x, +y, and +z directions.
Using unit vectors i and j, vector A is expressed as: A = Axi + Ayj
PHY 1151 Principles of Physics I

11. Adding Vectors Using the Components Method
Suppose that A = Axi + Ayj and B = Bxi + Byj.
Then, the resultant vector R = A + B = (Ax + Bx)i + (Ay + By)j.
When using the components method to add vectors, all we do is find the x and y components of each vector and then add the x and y components separately.
PHY 1151 Principles of Physics I

12. Example: The Sum of Two Vectors (with Components Method)
Two vectors A and B lie in the xy plane and are given by A = (2.0i + 2.0j) m and B = (2.0i - 4.0j) m.
(1) Find the sum of A and B expressed in terms of unit vectors.
(2) Find the x and y components of the sum.
(3) Find the magnitude R and direction  of the the sum.
PHY 1151 Principles of Physics I

13. Example: Adding Vectors Using Components
A commuter airplane takes a route shown in the figure. First, it flies from the origin of the coordinate system shown to city A, located 175 km in a direction 30.0° north of east. Next, it flies 153 km 20.0° west of north to city B. Finally, it flies 195 km due west to city C.
Find the location of city C relative to the origin. o
PHY 1151 Principles of Physics I

14. PHY 1151 Principles of Physics I
Homework
See online homework assignment at
Hand-written homework assignment:
Chapter 3, Page 77, Problems: #20
Note: When drawing the vectors, include the scale ratio used.
PHY 1151 Principles of Physics I