In order to determine how a system evolves in time, we first must define what our system is. The importance of defining your system will become more apparent as we get further along in our studies this term, but it is advantageous to now start thinking about what defines our system we wish to study.

We will start by considering objects with infinitesimal size, yet finite mass, often referred to as point particles. These point particles are an abstraction of real physical objects, such as a car, bicycle, person, dog, etc… Since we are interested in tracking the motion of these point particles through space and time, we need to ask ourselves the question, which quantities are of importance? Color? My attitude at this moment? Taste? Mass? Length? Position? Velocity? Acceleration?...etc. Believe it or not, this is a fairly involved question, so at this moment we will unfortunately leave this question mostly unanswered because we will not develop all the tools necessary in this course to truly appreciate the answer, but I promise we are building the foundation on which to dive into these types of fundamental questions. The key point is, in physics, we work with physical quantities which are either vectors or scalars. For the latter part of this lecture, the physical quantities of interest are position, displacement, velocity, and acceleration.

Throughout this series of physics, we make the following definitions regarding vectors and scalars; A scalar is a quantity with magnitude only; A vector is a quantity with magnitude and direction.

LECTURE 02: SCALARS, VECTORS, and COORDINATE SYSTEMS

WARM UP: Energy has the same dimensions as a force multiplied by a distance. Force has the same dimensions as mass multiplied by an acceleration. What are the dimensions of energy?

Select LEARNING OBJECTIVES:

Develop the mathematical tools necessary to work with vector algebra equations.i.Demonstrate the ability to relate different representations, specifically physical and mathematical representations, in the context of vector algebra.

ii.

Understand the directional relationship between two vectors when the vectors are related to each other by a scalar.

iii.

Convert between component notation and magnitude/direction notation. iv.

TEXTBOOK CHAPTERS:

Giancoli (Physics Principles with Applications 7th) :: 3-1 ; 3-2 ; 3-3 ; 3-4 •Knight (College Physics : A strategic approach 3rd) :: 1.5 ; 3.1 ; 3.3•BoxSand :: Math Tools ( Vectors ) •

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Some examples of scalars are; mass, time, distance, speed, area, volume… Note that scalars are mathematically described by one real number.

Some examples of vectors are; position, displacement, velocity, acceleration, force, momentum, impulse… Note that vectors are mathematically described by a set of coordinate specific numbers. To illustrate a vector in writing we will use the following notation:

Let's begin our study of vectors by exploring some formal statements. These statements of vectors will be demonstrated in the Cartesian coordinate system, which is your familiar x, y, and z-axis. To help visualize how the 3-D coordinate system looks, look at a bottom corner of the room you're in now, the edges of the walls form the axis, and you would be standing/sitting in one of the octants of a Cartesian coordinate system. The summary of this is found in the "CARTESIAN COORDINATE" page. Note, I will use 3-D <x,y,z> to illustrate how vectors mathematically follow the formal statements, but I will use 2-D <x,y> vectors when I wish to illustrate the pictorial representation of the statements (It gets messy when drawing in 3-D so I wish to avoid that).

One last thing before we look at the statements. Vectors can be visualized by "arrows" in space. Also, there are numerous ways to mathematically write a vector such as…

Equality of vectors1.

… all of which are equivalent to one another. I will always attempt to use the first form. Forgive me if I sometimes revert back to old habits and use one of the other forms, but always remember they are just different representations of the exact same thing, namely, in this case any arbitrary vector labeled A.

** Side note: We are using a fairly basic definition of what a vector is. Vectors are not entirely defined by our definition above. A more rigorous way is to define them based on how they behave under coordinate transformations. The arrows in space model will be suffice for all of our studies, but I just wanted to give you an insight into the more mathematical world out there.

**NOTE** For completeness, I have included all of the formal vector statements in this lecture, but we will not use some of the statements until later on this term and even some are left for next term. The statements you should familiarize yourself with are I through IX (i.e. don't worry about unit vector, dot product, or cross product yet; we will not cover/test you on any of those until the need for them arises. For example, when we get to the topic "work" this term, we will have to include the dot product to our set of tools associated with vectors.)

Note the angled brackets, they are angled to differentiate between a point in space that is usually mathematically described by a set of scalars with parentheses around them (x,y,z).

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Vector additionII.

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Multiplication by a scalarIII.

Vector subtractionIV.

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The null vectorV.

The Commutative Law of AdditionVI.

The Associative Law of AdditionVII.

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The Distributive LawVIII.

Magnitude of a vectorIX.

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Unit Coordinate Vectors (Basis Vectors)X.

The Scalar Product (Dot Product)XI.

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The Cross ProductXII.

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There are still a few more interesting properties of these vector statements to cover for Cartesian coordinates, along with some vector calculus. But with the above information, we can move forward and start working within the domain of kinematics.

I lied, before we move forward and start looking at some problems, we should first consider other coordinate systems. But you might be asking why? The Cartesian coordinates have probably never failed you in the past, nor will they fail you in the future, so why look at other coordinate systems? Well, "things" (things = notation, and pure length of writing) get messy sometimes. Thus, physicist are very well trained in the art of looking for symmetries. If the physical system you are modeling has some symmetries, (ex: constant radius), then there are other coordinate systems you can use which clean "things" up.

For now, I will only cover "Plane Polar Coordinates" in some detail. We will tackle "Cylindrical Coordinates" and "Spherical Coordinates" when we find the need to use them later on. A summary of plane polar coordinates can be found in the PLANE POLAR COORDINATES page. In this lecture, we will look at how the components of a vector in Cartesian coordinates relate to the components in plane polar coordinates.

Plane polar coordinates describes a vector by a radius and azimuthal angle in 2-D, hence the "plane" in

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Plane polar coordinates describes a vector by a radius and azimuthal angle in 2-D, hence the "plane" in plane polar coordinates. More commonly, it is just referred to as "polar coordinates". So from here on out, I will join the masses and drop the "plane". Shown below is how polar coordinates relate to Cartesian coordinates.

Please note the red text above! We will be working in Cartesian coordinates for the vast majority of PH201, so mistakes made by improper use of vector operations while in polar coordinates should be a minimum. But when in doubt, convert everything to Cartesian, do your vector addition and subtraction while in Cartesian coordinates, then convert back to polar coordinates if needed.

EXAMPLE [i, ii, iv] : The magnitude of a position vector is 10 m, and the angle from the positive x-axis to the positive y-axis is 30°. Write the position vector in Cartesian coordinates.

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EXAMPLE [i, ii, iv] : Find the magnitude and direction of the position vector,

PRACTICE [i, ii, iv] : You ride a bike for 15 miles at an angle of 25° from the (+)x-axis to the (-)y-axis. Find the x and y components.

PRACTICE [i, ii, iv] : A car is traveling with a velocity, . Find the magnitude (speed) and direction.

PRACTICE [i, ii] : Start at the corner of a cube, of side length 1 m, and find the distance to the furthest point that still resides on the cube.

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PRACTICE [i, ii] : Which of the following vector equations correctly describes the relationship among the vectors shown in the figure?

(1)

(2)

(3)

(4)None of the above equations are correct.(5)

PRACTICE [i, ii] : You take off from Mcnary (KSLE) airport in a small Cessna airplane. First you fly over Independence St. (7S5) airport. Next, you fly over Lebanon State (S30) airport. Then you fly over Albany (S12) airport. Finally you land at Corvallis (KCVO) airport. The map below shows the displacement vector for each leg of the flight.

Suppose that instead of flying (KSLE to 7S5) -> (7S5 to S30) -> (S30 to S12) -> (S12 to KCVO), you had followed the same vectors in a different sequence while still starting at KSLE: (S12 to KCVO) -> (7S5 to S30) -> (KSLE to 7S5) -> (S30 to S212). Mark on the map the final location of this flight sequence. If the

PRACTICE [i, iii] : If and .

Calculate the following: 2

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S30) -> (KSLE to 7S5) -> (S30 to S212). Mark on the map the final location of this flight sequence. If the distance from (KCVO to S12) is 15 miles, approximately how far away are you from the original destination, KCVO, after following this alternative sequence?

PRACTICE [I, ii] : Three wolves are pulling on the carcass of a dead animal but the net force on the animal is zero. The first wolf pulls with a force of N. The second wolf pulls with a force of magnitude equal to 3.162 N in a direction 18.435 degrees from the positive y direction towards the negative x direction. What must be the force of the third wolf?

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Conceptual questions for discussion

Is it possible to add two vectors of unequal magnitude and get zero for the magnitude of the resulting vector? Is it possible to add three vectors of equal magnitude and get zero for the magnitude of the resulting vector?

1)

Two forces are added using the head-to-tail method because forces are vector quantities.a.Force is a vector quantity because two forces are added using the head-to-tail method.b.

Which of the following two statements is more appropriate?2)

The value of a scalar.a.A vector.b.Component of a vector.c.The magnitude of a vector.d.

A situation may be described by using different sets of coordinate axes having different orientations (e.g. rotating a horizontal and vertical axes by some random angle). Which of the following do not depend on the orientation of the axes?

3)

is always greater than

Let 4)

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is always greater than a.

It is possible to have and b.

is always equal to c.

is never equal to d.

Is it possible to add a scalar to a vector? If so, demonstrate. If not, explain.5)

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