1. Chapter 3
Vectors in Physics

2. Units of Chapter 3 Scalars Versus Vectors The Components of a Vector
Adding and Subtracting Vectors
Unit Vectors
Position, Displacement, Velocity, and Acceleration Vectors
Relative Motion

3. Scalars Versus Vectors
Scalar: number with units
Vector: quantity with magnitude and direction
How to get to the library: need to know how far and which way

4. “As the crow flies”
Even though you know how far and in which direction the library is, you may not be able to walk there in a straight line:

5. The Components of a Vector
Can resolve vector into perpendicular components using a two-dimensional coordinate system:

6. Use some trig
Length, angle, and components can be calculated from each other using trigonometry:

7. Other Angles
What happens when the angle is not measured from the positive x – axis?

8. Signs of Components

9. TAPPS problem
Find the signs of the x- and y- components of vectors A, B, C, & D
Find the x- and y- components of vectors A, B, C, & D

10. From components to vectors

11. 3-3 Adding and Subtracting Vectors
Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last.

12. Moving Vectors
As long as you don’t change the length or the direction, you can move a vector anywhere and it does not change it.

13. Subtracting Vectors
The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, D = A – B.

14. Group Exercise
Consider the vectors A and B shown. Which of the other 4 vectors best represents:
A + B
A – B
B – A

15. 3-3 Adding and Subtracting Vectors
Adding Vectors Using Components:
Find the components of each vector to be added.
Add the x- and y-components separately.
Find the resultant vector.

16. Here is how it looks

17. 3-4 Unit Vectors
Unit vectors are dimensionless vectors of unit length.

18. 3-4 Unit Vectors
Multiplying vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

19. Group Exercise Write the vectors A, B, C & D in unit vector notation
Find the sum A+B+C in unit vector notation

20. 3-5 Position, Displacement, Velocity, and Acceleration Vectors
Position vector r points from the origin to the location in question.
The displacement vector Δr points from the original position to the final position.

21. 3-5 Position, Displacement, Velocity, and Acceleration Vectors
Average velocity vector:
(3-3)
So vav is in the same
direction as Δr.

22. 3-5 Position, Displacement, Velocity, and Acceleration Vectors
Instantaneous velocity vector is tangent to the path:

23. 3-5 Position, Displacement, Velocity, and Acceleration Vectors
Average acceleration vector is in the direction of the change in velocity:

24. ConcepTest What controls in your car cause you to accelerate?
The gas pedal
The brake
The steering wheel
1 and 2 only
1, 2 and 3

25. Circular Motion

26. 3-5 Position, Displacement, Velocity, and Acceleration Vectors
Velocity vector is always in the direction of motion; acceleration vector can point anywhere:

29. This illustration shows the path of a robotic vehicle, or rover
This illustration shows the path of a robotic vehicle, or rover. What is the direction of the rover’s average acceleration vector for the time interval from t = 0.0 s to t = 2.0 s?
A. up and to the left
B. up and to the right
C. down and to the left
D. down and to the right
E. none of the above

30. This illustration shows the path of a robotic vehicle, or rover
This illustration shows the path of a robotic vehicle, or rover. What is the direction of the rover’s average acceleration vector for the time interval from t = 0.0 s to t = 2.0 s?
A. up and to the left
B. up and to the right
C. down and to the left
D. down and to the right
E. none of the above

31. 3-6 Relative Motion
The speed of the passenger with respect to the ground depends on the relative directions of the passenger’s and train’s speeds:

32. Summary of Chapter 3 Scalar: number, with appropriate units
Vector: quantity with magnitude and direction
Vector components: Ax = A cos θ, By = B sin θ
Magnitude: A = (Ax2 + Ay2)1/2
Direction: θ = tan-1 (Ay / Ax)
Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

33. Summary of Chapter 3
Component method: add components of individual vectors, then find magnitude and direction
Unit vectors are dimensionless and of unit length
Position vector points from origin to location
Displacement vector points from original position to final position
Velocity vector points in direction of motion
Acceleration vector points in direction of change of motion
Relative motion: v13 = v12 + v23