In this packet we will be taking a look: Defining vectors and scalars Relating vectors and scalars to one

another Examples of vectors and scalars Adding and Subtracting vectors Usefulness of vectors and scalars

Note: In this packet we will be discussing vectors and scalars as they relate to physics. Depending on the subject you are studying the definition may change slightly, but the information contained herein should still be useful for a general understanding.

Definitions:

Vector: A quantity which has both a magnitude and direction.

Scalar: A quantity which only has a magnitude.

Examples of a vector include displacement, velocity, and acceleration. Anything which has some measured value and a direction can be thought of as a vector.

Examples of a scalar include speed, mass, volume, and density. Anything which just has some measured value but no direction can be thought of as a scalar.

Let’s start with scalars since it is the easier of the two and the one you have been using without even thinking about it.

A scalar is represented by just a number:

Example 1:Q: How old are you?A: Because the answer to this question is just a number we call it a scalar and represent the scalar by just writing down the number.

Sample Problems:1.How many fingers do you have?2.How long did it take you to finish your homework?3.What is the distance from Minneapolis, Minnesota to New York City, New York?

Notice how we do not have to worry about any sort of direction when answering these questions. Our responses might be 10 fingers, 3 hours, and 2500 miles, respectively, but they are all quantities without a direction.

Now let’s look at how a vector is represented. By definition we know that a vector has to have a magnitude (a measureable value) and direction.

Let’s go back to the last question in the sample problems. We can also represent the displacement from Minneapolis, MN to New York, NY by a vector.

To travel from Minneapolis to New York we would have to travel approximately 2500 miles, but we would have to travel East.

Here 2500 miles is our magnitude and East is our direction, so we have a vector or some displacement.

Notice how the distance represents a scalar and displacement represents a vector in this case. The student should be aware that displacement does not just apply to length, but for most physics problems you should be alright.

N

Sample Questions: (Stating a general direction N,S,E,W is enough)1.What is the displacement from New York City, New York to Minneapolis, MN.2.What is the displacement from New Orleans, Louisiana to Minneapolis, MN.3.What is the displacement from Madrid, Spain to London, England?

With this we have only grazed the surface of vectors. Another major distinction which understanding vectors and scalars helps you make is between velocity and speed.

If we were in a car on the highway and you asked me how fast I was driving I would say 55 mph, which would be a scalar. This scalar is represented by speed.

If you asked me what my heading was, I might say 55 mph North. This vector is represented by velocity.

Definitions:

Displacement: Some measurable quantity (in the example’s it is the distance from original starting point) and direction.

Velocity: A quantity possessing both speed and direction. For physics purposes you should think of it as displacement divided by some amount of time.

Another important example we will look at for a vector quantity in this packet is acceleration. Acceleration measures how velocity changes with time. Notice that since acceleration is based on a vector, velocity, it is also a vector.

Example 2:What is the acceleration of an apple as it is falling to Earth?

You might say well since acceleration on the Earth is 9.81 m/s2 that the acceleration of the apple is just that. But doesn’t the apple have a direction? Indeed the apple is falling to the ground from a tree so we say the direction of the apple is downward towards the center of the Earth. Thus the acceleration has a magnitude of 9.81 m/s2 directed downward.

Example 3:How much do you weigh? What is your mass?

Q: What is the difference between weight and mass?

A: Weight is dependent on gravity (or acceleration due to the Earth) while mass does not.

If you were to weigh yourself on a bathroom scale you would get some quantity to show up on the scale but does that quantity have a direction? Here as with the apple example your weight is directed downward towards the center of the Earth.

If you were to divided the number you got for your weight by 9.81 m/s2, if you’re measuring in kilograms, or 32 ft/s2, if you’re measuring in pounds, you will get your mass.

Q: Which of these quantities is a vector and or scalar?

A: Weight is a vector and mass is a scalar.

To understand vectors a very common tool which we use is the coordinated plane or the x and y-axis. I will be using the coordinate plane to give a geometric representation of a vector and then give it an algebraic interpretation.y + or

N

x + or E

A

B

(0,0)

(3,4) Usually when we begin to consider vectors we set the origin of the coordinate plane to represent our starting point and plot our ending point appropriately. In this case say I started a trip from my house and travelled northeast to the mall. The mall can be thought of as being 3 miles east and 4 miles north or, 5 miles northeast from home.We say the vector AB is 5

miles northeast. -or- northeastmilesAB 5

The student should notice from the previous picture and the symbol representation of the vector that the arrow above AB indicates a vector and it means from point A to point B. Sometimes a vector is also indicated by bold letters with an arrow or just ^ above a letter, meaning a unit vector.

The student should also take notice that I said the vector can be represented in two ways, 3 miles east and 4 miles north or, 5 miles northeast.

The 5 miles northeast represents the vector and 3 miles east and 4 miles north represent what we call the components of the vector. Components can be thought of as the building blocks of vectors on the coordinate plane.

Whenever we draw a vector between two points we can usually draw a right triangle between those points as shown on the previous picture. The hypotenuse of the triangle can be thought of as the vector and the sides of the triangle are the components of the vector.

Last but not least the length of the vector indicated the magnitude, so a longer line means a bigger magnitude and a shorter line means a smaller magnitude.

Algebraically we would represent the same information contained in the previous graph as follows:

yxAB ˆ4ˆ3

This means the vector from A to B is 3 units in the positive x-direction and 4 units in the positive y-direction.

Now you may ask why we would want to represent the vector in its component parts. The reason we do this is because sometimes we have more than one vector to worry about and we may need to add them together to figure out say our total displacement from our original starting point, but if we didn’t know the component parts this can sometimes be hard to do. But, if we have the component parts we can easily add up all the x-displacements together to get the total x-displacement and add up all the y-displacements together to get the total y-displacement.

We usually say we must add corresponding components of vectors together and this is what is meant by that. The reason being that x and y are perpendicular to one another so their corresponding components cannot be added to one another.

Now that we have covered breaking vectors into components we can look at an example to see how we can add and subtract vectors from one another.

Example 3:N

(2,4)(-

1,2)

(0,0)

A

B

C

Suppose I went from my home, point A, to the mall, point B, and then to the movie theater, point C. What was my total displacement?

Now remembering that in order to calculate the vector AC (total displacement) we need to find the vectors AB and BC and get their components and add them together.Q: What are the components of AB? BC? What is vector

AB/BC/AC? Why is my total displacement not the total distance I travelled?

A: Displacement is a vector and distance is a scalar. Total displacement tells where I started from to where I currently am.

Now for vector AB we know that it is 2 units in the positive x-direction and 4 units in the positive y-direction from A. Vector BC is 3 units in the negative x-direction and 2 units in the negative y-direction from B. We also know that vector AC is 1 unit in the negative x-direction and 2 units in the positive y-direction. Writing this information down algebraically we get the following:

yxAC

yxBC

yxAB

ˆ2ˆ1

ˆ2ˆ3

ˆ4ˆ2

If we want to find our total displacement from point A to point C we add the components of vectors AB and BC together to give us:

yxBCAB

yxBCAB

ˆ2ˆ1

ˆ24ˆ32

The student should note that this vector exactly corresponds with what we get for the vector AC, which in this case helps us to check our answer.