Before you can build a bridge or even think about completing a design, you have to begin by understanding the way engineers depict the world around them, a fundamental concept in statics. Enter the vector. Essentially, the study of vectors is the first step into this larger world of statics analysis.
This chapter focuses on exploring the behavior of vectors, seeing the commonalities in their construction, and understanding some of the subtle differences in their creation and application. In this chapter, I define the three major pieces of information you use to help a vector properly describe an action, I show you a few ways to draw a vector, and I break down the three primary types of vectors. This chapter won't have you building a bridge immediately, but it does help you take your first step in getting a handle on the world around you in proper statics style.
You quickly discover that the ability to create and define a proper vector is an invaluable set of skills. This ability lends itself fluidly to solving statics problems. However, before you can become truly proficient in statics, you first need to understand some basic terminology and the three pieces of information you need to properly define a vector.
Simply put, a vector is a quantity that helps describe the way that an action is applied to an object or group of objects. For example, a velocity vector can describe the velocity motion of a golf ball after it has been hit by a nine-iron, and a distance vector can help depict how far away and in what direction it landed. A force vector can describe how hard and in what direction the golf club strikes the golf ball.
Many different types of vectors exist — from velocity and displacement vectors, to vectors that describe magnetic field behavior, to vectors that are mathematical solutions of differential equations. In statics, the force vector is the main type of vector you have to deal with. (Note: Don't confuse these types of vectors with the categories of vectors I describe later in the chapter.)
Before you can dive very far into the heart of your study in statics, you need to understand the difference between a vector and a scalar, two terms that are always popping up in statics textbooks and practice:
Scalar: A scalar quantity (or simply a scalar) is any measurement made only with regard to an action's amount (its magnitude) and not its direction. Examples of scalar quantities include the cost of this book, the temperature of the room around you, or the airspeed of Monty Python's unladen swallow. You can describe all these quantities as a single amount. Even time is considered a scalar quantity because time only moves in one direction (supposedly).
Vector: As I mention earlier in the chapter, a vector is a quantity that describes both the size (an amount) and direction of a particular action. Examples of a vector include the approach flight path of an airplane coming in for a landing (the distance from the runway is a scalar quantity, and the flight path trajectory defines the direction), the velocity of a speeding car (the speedometer reading is a scalar, and the compass on the dashboard indicates the direction), or the force of an elephant sitting on a chair (the mass of the elephant is a scalar quantity, and the direction of gravity defines the direction of the elephant's force).
In fact, a scalar quantity is often part of the information contained within a vector definition, but I talk more about that in Chapter 5. For example, the speed of a moving elephant (a scalar quantity) is directly related to the velocity vector of that mammal.
Although you can display a scalar entity simply by jotting a number or measurement on your piece of paper, the proper representation of a vector requires three pieces of very specific information:
Magnitude: The magnitude is the numerical value of a given vector. Constructing a vector requires actually knowing a scalar quantity — the magnitude. Magnitudes of vectors are scalar quantities and may be positive, negative, or zero in value. Just remember that there is no direction associated with a magnitude. (Sit tight — I talk about how to actually create a vector from a scalar quantity in Chapter 5 and how to calculate the magnitude of a vector in Chapter 8.)
Sense: The sense of a vector is the sign of the magnitude, or the direction in which the vector is acting. The sense is the part of a vector that indicates whether a charging elephant, moving at a speed of 20 miles per hour, is heading toward or away from you. You can describe that direction in several ways, and I cover each of them in Chapter 5.
Point of application and lines of action: The point of application is the physical location on the object or in space where the vector is acting. The line of action of a vector refers to the line in space on which the vector is acting, regardless of whether the vector is acting internally or externally to the object. In all cases, however, the line of action of a vector and the vector's point of application (if it has one — certain types of vectors don't!) always coincide. That is, the line of action of a vector passes through the point of application.
Note
Vectors can act either internally or externally:
External vector: A vector that acts on the external surface of an object. Examples of external vectors include drag forces on the wing of an airplane, the friction forces you feel when you rub the palms of your hands together, and the force of your hands on the cover of this book as you read it. (I discuss external vectors in more detail beginning in Chapter 7.)
Internal vector: A vector that acts on the object at a specific internal location. Examples of internal vectors include the weight of this book and the internal compressive force in the legs of the chair you're sitting in (if you're sitting). I dive deeply into the subject of internal vectors beginning in Chapter 16.
Knowing whether vectors are acting externally or internally can help you decide how to best tackle a given problem.
Sometimes, there's no substitute for a good example. In this section, I outline a scenario that I hope helps you get a firm grasp on vectors and their components (which I cover in the preceding section). Imagine trying to give driving directions to a friend travelling from his house to yours. In your discussion with him, you'd never have a list of directions that states
"Okay, Tom, first go a half of a mile, and then go another three-quarters of a mile, and finally head a quarter of a mile."
The representation of this directional data is an example of a scalar quantity. All three distances in the previous statement are nonnegative values and are actually all considered magnitudes.
Unless you simply don't want visitors or you like getting frantic cellphone calls from lost friends (and admit it, who doesn't?), you need to provide significant information that's missing from that first list of directions.
In this case, you definitely need to describe the sense, or the direction, of each of those measurements. You can establish the sense of these directions by using relative descriptions such as "turn left" or "veer right," but this approach can be dangerous. If your intended direction is west and you tell someone accidentally heading south to turn left, his final direction will be completely opposite of where you want him to go. One wrong turn can render relative descriptions completely inaccurate. To avoid this dilemma, use absolute sense descriptions by giving cardinal directions with the proper instructions. As you begin to construct vectors, take special care to formulate your vectors with specific absolute information.
In Tom's case, a better set of the previous driving directions, incorporating absolute sense description, may be
"Okay, Tom, first go a half of a mile north, and then go another three-quarters of a mile east, and finally head a quarter of a mile south."
However, even this set of instructions is still lacking a significant piece of specific information. In this case, though the directions themselves are decent and definitely more detailed, you still don't know where the trip starts. Tom's starting point in this example represents the point of application.
The current instructions are adequate for relative positioning, although the final destination becomes directly dependent on the starting point — if you change the starting point, the final destination obviously changes as a result. You can vastly improve these relative directions if you also mention the starting position:
"Okay, Tom, from your home, first go a half of a mile north, and then go another three-quarters of a mile east, and finally head a quarter of a mile south. This will get you to my house."
From this list of directions, your visitor should have very little trouble moving from his house to yours (as long as he doesn't hit any construction zones or detours along the way)!
Unfortunately, in describing driving directions for how your friend travels from his house to yours, you don't usually see the path as a line on the ground or on the map (unless of course his car has an oil leak). Similarly, you can't physically see a force vector in action; you see only its resulting influence on the object. Because engineers are always making sketches to help describe the world around them, you need to be familiar with the techniques they use to graphically depict a vector.
To draw a vector, you have to graphically represent the three major components of a vector: magnitude, sense, and point of application, which I discuss in the preceding section. In the sections that follow, I explain how you graphically represent this information when drawing a vector, as well as describe two common vector depictions.
In this book, as in most statics and mathematics references, I typically represent vectors as single-headed arrows that break down into a number of parts (which you can see in Figure 4-1):
Head: The arrowhead indicates the vector's sense. However, an exception to this guideline can occur when vectors are pushing on an object; in this case, the head of the arrow is commonly used to indicate the point of application. I explain this exception in more detail in Chapter 9.
Tail: The tail of the arrow typically depicts the vector's point of application (barring the exception described in the preceding bullet).
Shaft: The actual line-length of the arrow represents the vector's magnitude — a longer vector drawing implies a larger action and vice versa. The shaft of the arrow aligns with the axis or line of action of the vector. Vectors that aren't oriented horizontally or vertically sometimes include an angle measurement to help define the vector's orientation. This angle is usually measured from either a horizontal or vertical reference. I discuss vector notation more in Chapter 5.
Label: The label of the vector can be a letter or name given to a particular vector arrow to help distinguish it from other vectors in your drawings. Sometimes, you actually write the value (with proper units, of course) of the magnitude of the vector as the label. In other situations, you may use an alphanumeric label such as Load1, WindForce, or Bob'sWeight — the sky is the limit on how you actually label your vectors, but it helps if you name it something to remind you of what that arrow represents on the picture. To help distinguish these entities, some textbooks (and this book) write nonnumeric vector labels in bold: V. Other texts commonly display vector labels with an arrow symbol over the top:
This label serves two basic purposes:
It acts as a reminder of specific scalar information that you may have calculated previously or already know from a statement given in the problem. If you don't know this specific piece of information at the time (and you often don't), you can use the name or label of this vector as variables in your equations.
For the sake of convenience, you often don't include the magnitude of the vector graphically (by making a vector longer or shorter) in your depiction. Instead, you write the numerical quantity of the magnitude (if it's known) beside the arrow. Doing so saves space when you're constructing free-body diagrams (which I dive into in Part IV) and helps really small vectors remain visible. The drawback: You lose the ability to perform graphical computation techniques, so you have to rely on vector equations and basic geometry to complete your calculations (which I discuss in Chapter 7).
Beginning in Part V, I show you several different techniques for solving for unknown vector magnitudes. Remember that regardless of which method you use to depict the magnitude, the proper representation of a vector always includes the vector's sense. I delve into the ways you can represent a vector's sense in Chapter 5.
Another useful notation is the double-headed arrow, which helps depict the rotation of an action such as the turn of a doorknob (see Figure 4-2). The double- headed vector contains much the same information as the single-headed variety. Specifically, the tail, shaft, and label designations of a double-headed vector are all similar to their single-headed cousins described in the preceding section, except this version has two heads.
Single- and double-headed notation are relatively similar; the difference between this illustration and Figure 4-1 is that it has two arrowheads (hence the name) and replaces the line of action with an axis of rotation. These changes show that the diagram describes a rotation behavior. The double-headed vector provides some liberties in how you perform calculations with vectors that describe a rotation, which I discuss in more detail in Chapter 12.
Most vectors fall into one of three categories, all of which have similar requirements — namely, that they must have a magnitude and a sense. However, not all three categories have a specific point of application. For example, in the case of the sliding vector, the point of action is replaced by a line of action. (Check out the earlier "Defining a Vector" section for more on the required properties of vectors.) Table 4-1 provides you with a snapshot of requirements of the three main vector categories I cover in this section. In the following sections, I explain the different categories of vectors and give examples for each.
Table 4.1. Types of Vectors and Their Requirements
Type of Vector | Magnitude | Sense | Point of Action |
|---|---|---|---|
Free | Yes | Yes | No |
Fixed | Yes | Yes | Yes |
Sliding | Yes | Yes | Not exactly |
A fixed vector is a type of vector where the point of application is set at a distinct location and can't be moved without changing the behavior of the initial problem. Examples of fixed vectors include the velocity of a particle and the gravitational weight of a rigid body.
Figure 4-3 illustrates two different fixed vectors. In Figure 4-3a, the fixed vector represents the gravitational influence (or self weight) of the object. The center of gravity occurs at only one location within an object. (I discuss the center of gravity concept in further detail in Chapter 10.) In Figure 4-3b, the fixed vector represents a velocity vector and indicates the speed and direction of the object. The velocity vector illustrates the behavior of the particle on which it's acting.
Another type of vector is the free vector, which doesn't necessarily have a specific point of application but rather acts more generally on an object. These vectors, such as moments and couples (which you can read about in Chapter 11), result in a specific action but may be freely moved around the object without changing the original behavior. Figure 4-4 shows you an example of a free vector. Note how the object rotates with the same intensity (or magnitude) and direction in space regardless of where the action is applied on the object.
A third type of vector is the sliding vector, which is also sometimes referred to as a line vector. Sliding vectors (such as forces on rigid bodies) may freely move on an object as long as they remain on their line of action. Unlike the fixed vector, the sliding vector doesn't have a distinct point of application but rather acts more generally along a specific direction line (as you can see in Figure 4-5).
In fact, this sliding notion leads to the principle of transmissibility, which is one of the major requirements in the study of rigid body statics (I discuss this principle more in Chapter 9). In statics, a large number of the vectors you deal with are considered sliding vectors, or vectors that have lines of action. Sliding vectors have very useful properties: They can be moved anywhere along their lines of action and still maintain their original behavior. Part III covers these properties in more detail.
