1. Physics/Physical Science
What is a vector?
Physics/Physical Science

2. A study of motion will involve the introduction of a variety of quantities which are used to describe the physical world.
Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories - vectors and scalars.

3. Vectors vs. Scalars The difference between scalars and vectors
If a quantity has only a size, it is called a scalar. Time and temperature are examples of scalars. Mass, too, is an example of a scalar.
If a quantity has a size and a direction, it is called a vector and can be symbolized, or drawn, as an arrow. Velocity is an example of a vector.

4. What is a Vector?
A vector quantity is a quantity which is fully described by both magnitude and direction.
The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces which occur in two dimensions.

5. Examples of vector quantities which have been previously discussed include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed.

6. Do Not Write
For example, suppose your teacher tells you "A bag of gold is located outside the classroom. To find it, displace yourself 20 meters." This statement may provide yourself enough information to pique your interest; yet, there is not enough information included in the statement to find the bag of gold.

7. Do Not Write
The displacement required to find the bag of gold has not been fully described. On the other hand, suppose your teacher tells you "A bag of gold is located outside the classroom. To find it, displace yourself from the center of the classroom door 20 meters in a direction 30 degrees to the west of north."

8. Do Not Write
This statement now provides a complete description of the displacement vector - it lists both magnitude (20 meters) and direction (30 degrees to the west of north) relative to a reference or starting position (the center of the classroom door). Vector quantities are not fully described unless both magnitude and direction are listed.

9. Displacement
To test your understanding of this distinction, consider the motion depicted in the diagram below. A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North.

10. Even though the physics teacher has walked a total distance of 12 meters, her displacement is 0 meters. During the course of her motion, she has "covered 12 meters of ground" (distance = 12 m). Yet when she is finished walking, she is not "out of place" - i.e., there is no displacement for her motion (displacement = 0 m).

11. Displacement, being a vector quantity, must give attention to direction. The 4 meters east is canceled by the 4 meters west; and the 2 meters south is canceled by the 2 meters north. Vector quantities such as displacement are direction aware.

12. Vector quantities are often represented by scaled vector diagrams
Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object.

13. Criteria of a vector diagram:
An example of a scaled vector diagram is shown below. Observe that there are several characteristics of this diagram which make it an appropriately drawn displacement vector diagram.
Criteria of a vector diagram:
a scale is clearly listed
a vector arrow (with arrowhead) showing a specified direction.
the magnitude of 20 m and the direction is (30 degrees West of North). .
                                 

14. The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale.

15. For example, the diagram below shows a vector with a magnitude of 20 miles. Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector arrow is drawn with a length of 4 cm. That is, 4 cm x (5 miles/1 cm) = 20 miles.

16. In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated (such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction

17. Practice Problems

18. What is the magnitude and direction?

19. Vector Addition
A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant).

20. Recall in our discussion of Newton's laws of motion, that the net force experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object. That is the net force was the result (or resultant) of adding up all the force vectors.

21. Observe the following summations of two force vectors:
                                                                              

22. These rules for summing vectors were applied to free-body diagrams in order to determine the net force (i.e., the vector sum of all the individual forces). Sample applications are shown in the diagram below.

23. The task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. For example, a vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below.

24. Resultants
The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R. As shown in the diagram, vector R can be determined by the use of an accurately drawn, scaled, vector addition diagram.

25. Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component.

26. If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component.

27. If the single chain were replaced by two chains
If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference.

28. What are the components of the picture?

29. The process of determining the magnitude of a vector is known as vector resolution. The two methods of vector resolution which we will examine are:
the parallelogram method
the trigonometric method

30. The two vectors at right angles to each other, that add up to a given vector are known as the components (Ax, & Ay).

31. The diagram shows that the vector is first drawn to scale in the indicated direction; a parallelogram is sketched about the vector; the components are labeled on the diagram; and the result of measuring the length of the vector components and converting to m/s using the scale. (NOTE: because different computer monitors have different resolutions, the actual length of the vector on your monitor may not be 5 cm.)

32. As the 60-Newton tension force acts upward and rightward on Fido at an angle of 40 degrees, the components of this force can be determined using trigonometric functions.

33. What is a projectile?
A projectile is an object upon which the only force acting is gravity.
A projectile is any object which once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

35. Thus, the free-body diagram of a projectile would show a single force acting downwards and labeled force of gravity (or simply Fgrav). Regardless of whether a projectile is moving downwards, upwards, upwards and rightwards, or downwards and leftwards, the free-body diagram of the projectile is still as depicted in the diagram at the right

36. Their belief is that forces cause motion; and if there is an upward motion then there must be an upward force. They reason, "How in the world can an object be moving upward if the only force acting upon it is gravity?" Such students do not believe in Newtonian physics

37. A force is not required to keep an object in motion
A force is not required to keep an object in motion. A force is only required to maintain an acceleration. And in the case of a projectile that is moving upward, there is a downward force and a downward acceleration. That is, the object is moving upward and slowing down.

38. Consider a cannonball shot horizontally from a very high cliff at a high speed. And suppose for a moment that the gravity switch could be turned off such that the cannonball would travel in the absence of gravity? What would the motion of such a cannonball be like? How could its motion be described?

40. If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon? To ponder this question, first consider a scenario in which there is no gravity acting on either the banana or the monkey. What would be the path of the banana? Would the banana hit the monkey?