Vectors and Their Applications

chapter 1

Elementary Operations

1-1 Scalars and Vectors

In discussing physical space it is necessary to consider several types of physical quantities. One class of quantities consists of those quantities which have associated with them some measure of undirected magnitude. Such quantities are called scalar quantities or simply scalars. Each scalar quantity can be represented by a real number which indicates the magnitude of the quantity according to some arbitrarily chosen convenient scale or unit of measure. Since scalars are real numbers, scalars enter into combinations according to the rules of the algebra of real numbers. Mass, density, area, volume, time, work, electrical charge, potential, temperature, and population are examples of scalar quantities.

A second class of physical quantities consists of those quantities which have associated with them both the property of magnitude and the property of direction. Such quantities are called vector quantities or simply vectors. Force, velocity, acceleration, and momentum are examples of vector quantities.

The following example illustrates the need to distinguish between scalars and vectors. Consider a plane which flies from point A to point B, 300 miles east of A, and then proceeds to fly north to a third point C, 400 miles north of B as shown in Figure 1 – 1. The distances which the plane has flown are scalars and may be added in the usual manner to determine the total distance covered by the flight; that is, 300 + 400, or 700, miles. The flight may also be considered in terms of the displacement of the plane from point A to point C. This displacement may be considered as the sum of two displacements: one 300 miles east from A to B and the second 400 miles north from B to C; the sum of these displacements is the displacement 500 miles with a bearing of approximately 36°52’ east of north from A to C. Notice that direction as well as magnitude is considered in describing displacements. Displacements are examples of vector quantities. Furthermore, the sum of two displacements is calculated in a rather different manner than the sum of two scalar quantities; in other words, vector addition is quite different from scalar addition.

Figure 1-1

Figure 1-2

Notice that in Figure 1-1 the displacements were denoted by means of directed line segments or arrows. In mathematics it is convenient to construct a geometric model of the physical concept of a vector and of situations involving vector quantities.

Definition 1-1 A geometric vector or simply a vector is a directed line segment (Figure 1-2).

In Figure 1-2 the length of the directed line segment with reference to some conveniently chosen unit of length is associated with the magnitude of the vector; thus, lengths of directed line segments represent scalars. Notice that the magnitude of a vector is a non-negative real number. Unless otherwise restricted, we shall consider a geometric vector as a vector in three-dimensional space.

The notation for a vector which we shall use is due to the mathematician Argand. Symbolically the vector represented in Figure 1-2 is denoted by e9780486148892_i0004.jpg , where A is the initial point (sometimes called the origin or origin point) of the directed line segment and B is the terminal point. Symbolically the magnitude of e9780486148892_i0005.jpg is denoted by | e9780486148892_i0006.jpg | . Whenever convenient, a second notation for vectors will be used which consists of single small letters beneath a half arrow such as e9780486148892_i0007.jpg , e9780486148892_i0008.jpg , e9780486148892_i0009.jpg , .... Then e9780486148892_i0010.jpg , e9780486148892_i0011.jpg , e9780486148892_i0012.jpg , . . . will represent the magnitudes of such vectors.

In our study of vectors we shall often associate a vector with a line segment. It is possible to associate either e9780486148892_i0013.jpg or e9780486148892_i0014.jpg with the line segment whose end points are A and B, where in each case the magnitude of the vector is equal to the length of the line segment. However, the vectors e9780486148892_i0015.jpg and e9780486148892_i0016.jpg are not equal. When we use the second notation of a single small letter to represent a vector associated with a line segment, we will find it convenient to adopt the following convention for establishing the association we desire: "associate e9780486148892_i0017.jpg with line segment AB" shall mean e9780486148892_i0018.jpg ; "associate e9780486148892_i0019.jpg with line segment BA" shall mean e9780486148892_i0020.jpg Note that line segments AB and BA are identical and thus that our convention is strictly a notational procedure for this book.

A vector of particular interest is the zero vector or null vector, which will be denoted by e9780486148892_i0021.jpg .

Definition 1-2 A null vector is a vector whose magnitude is zero.

By Definition 1-2, a geometric vector is a null vector if its initial and terminal points coincide. We choose to consider a null vector as a vector without a unique direction and, specifically, with a direction that is indeterminate. The null vector is the only vector whose direction is indeterminate. Some mathematicians choose to consider the null vector as having any arbitrary direction; that is, a null vector could be considered as the limit of any one of an infinite number of finite vectors as its magnitude approaches zero. The wording of many theorems in the subsequent development of vector algebra must be carefully changed if the direction of the null vector is considered arbitrary.

Exercises

Identify each quantity as either a scalar quantity or a vector quantity.

Distance between New York and Boston.

Displacement from Chicago to St. Louis.

Temperature of 97° Fahrenheit.

Weight of 100 pounds.

Pressure of 18 pounds per square inch.

250 horsepower.

1-2 Equality of Vectors

In choosing an appropriate definition for the equality of two vectors one is usually guided by the applications that will be made of the vectors. For example, in the study of the theory of mechanics of rigid bodies, vectors e9780486148892_i0022.jpg and e9780486148892_i0023.jpg (denoting forces) have the same mechanical effect in that the line of action of these two vectors is the same and the vectors have the same magnitude. However, as in Figure 1-3, the mechanical effect of e9780486148892_i0024.jpg , a vector of equal magnitude to e9780486148892_i0025.jpg and e9780486148892_i0026.jpg , would be to rotate the shaded object acted upon. In this type of problem e9780486148892_i0027.jpg and e9780486148892_i0028.jpg would be considered equal since they have equal magnitudes and lie along the same line with the same orientation. Consideration of this type of problem requires a definition of equality for a class of vectors commonly called line vectors.

Figure 1-3

In the theory of mechanics of deformable bodies one needs a more restrictive definition for the equality of vectors. For example, consider e9780486148892_i0030.jpg and e9780486148892_i0031.jpg , both having equal magnitudes and directed along the same line with the same orientation, acting upon an elastic material as indicated in Figure 1-4. Each vector would deform the material in a different way; e9780486148892_i0032.jpg would tend to compress it, while e9780486148892_i0033.jpg would tend to stretch it. A consideration of this type of problem leads to a definition for the equality of vectors: vectors with equal magnitudes and acting along the same line of action with the same orientation are to be applied at the same point in space. A study of vectors under this definition of equality is a study of bound vectors.

Figure 1-4

However, those properties of geometry and trigonometry with which one is generally interested in mathematics allow mathematicians to use a less restrictive definition for the equality of vectors.

Definition 1-3 Two vectors e9780486148892_i0035.jpg and e9780486148892_i0036.jpg are equal if, and only if, e9780486148892_i0037.jpg and e9780486148892_i0038.jpg is parallel to e9780486148892_i0039.jpg with the same orientation; that is, e9780486148892_i0040.jpg if, and only if, e9780486148892_i0041.jpg and e9780486148892_i0042.jpg have the same magnitude and direction.

In Definition 1-3 the word parallel is used in a generalized sense to mean that the vectors are on the same or parallel lines; orientation refers to the sense of the vector along the line; direction refers to both parallelism and orientation. From the definition it follows that any vector may be subjected to a parallel displacement without considering its magnitude or direction as being changed. The vectors in Figure 1-5 are all equal to one another. Since the vectors are equal, any one of the vectors may be considered to represent a whole class of equal vectors of which the others are members.

Under Definition 1-3 for the equality of vectors, the vectors discussed are called free vectors. Unless otherwise specified, we shall assume that all vectors are free vectors.

Figure 1-5

Exercises

In Exercises 1 through 6 state whether or not the two vectors appear to be equal.

In Exercises 7 through 9 copy the given figure and draw a vector with the given point P as its initial point and equal to the given vector.

10 – 12. Copy the given figure in Exercises 7 through 9 and draw a vector with the given point P as its terminal point and equal to the given vector.

1-3 Vector Addition and Subtraction

Consideration of the displacement problem of § 1-1 and similar physical problems concerning vectors motivated the following definition called the law of vector addition.

Definition 1-4 Given two vectors e9780486148892_i0046.jpg and e9780486148892_i0047.jpg , if e9780486148892_i0048.jpg is translated so that its initial point coincides with the terminal point of e9780486148892_i0049.jpg , then a third vector e9780486148892_i0050.jpg with the same initial point as e9780486148892_i0051.jpg and the same terminal point as e9780486148892_i0052.jpg is equal to e9780486148892_i0053.jpg (Figure 1-6).

The sum of two vectors is a uniquely determined vector; that is, if e9780486148892_i0054.jpg and e9780486148892_i0055.jpg , then e9780486148892_i0056.jpg .

Figure 1-6

Figure 1-7

Consider any parallelogram ABCD. Associate vectors e9780486148892_i0059.jpg and e9780486148892_i0060.jpg with sides AB and BC, respectively, as in Figure 1-7. Then e9780486148892_i0061.jpg may be associated with the