What is directed line in vector?

1.13.1 Linear, Circular, and Elliptic Polarization

As mentioned at the end of Section 1.12.3, the linearly polarized plane wave solution described in Section 1.12.2 corresponds to only one among several possible states of polarization of a monochromatic plane wave, where the term ‘state of polarization’ refers to the way the instantaneous electric and magnetic intensity vectors are related to the wave vector k.

If we consider, for concreteness, a plane wave propagating along the z-axis of a right-handed Cartesian coordinate system (for which n^, the unit vector along the direction of propagation, is ê3, the unit vector along the z-axis; we denote the unit vectors along the x- and y-axes as ê1and ê2), relations (1.55c) imply that the amplitude vectors E0 and H0 can point along any two mutually perpendicular directions in the x-y plane. One can assume, for instance, that these two point along ê1and ê2, respectively. This will then mean that the electric and magnetic intensity vectors at any point in space oscillate in phase with each other along the x- and y-axes.

More generally, a linearly polarized monochromatic plane wave propagating along the z-axis can have its electric vector oscillating along any other fixed direction in the x-y plane, in which case its magnetic vector will oscillate along a perpendicular direction in the same plane, where one has to keep in mind that for a progressive plane wave the electric vector, the magnetic vector, and the direction of propagation have to form a right-handed orthogonal triad—a requirement imposed by Maxwell’s equations. Thus one can think of a linearly polarized monochromatic plane wave propagating in the z-direction, where the directions of oscillation of the electric and magnetic intensities in the x-y plane are as shown in Fig. 1.2.

Such a linearly polarized wave can be regarded as a superposition of two constituent waves, each linearly polarized, the phase difference between the two waves being zero. More precisely, consider the following two plane waves, both with frequency ω and both propagating along the z-axis, and call these the x-polarized wave and the y-polarized wave, respectively:

(1.59a)(x-polarized wave)E1=ê1A1exp[i(kz−ωt)],H1=ê2A1μvexp[i(kz−ωt)],

(1.59b)(y-polarized wave)E2=ê2A2exp[i(kz−ωt)],H2=−ê1A2μvexp[i(kz−ωt)].

Here A1 and A2 are positive constants representing the amplitudes of oscillation of the electric intensities for the x- the y-waves. Evidently, these formulae represent linearly polarized waves, the first one with the vectors E and H oscillating along the x- and y-axes, respectively, and the second one with these vectors oscillating along the y- and x-axes, where in each case the instantaneous electric and magnetic intensities and the unit vector ê3form a right-handed orthogonal triad.

The superposition of these two waves with the same phase,

(1.60a)E=E1+E2,B=B1+B2,

then gives rise to the linearly polarized plane wave described by Eqs. (1.55a)–(1.55c), where now

(1.60b)n^=ê3,E0=ê1A1+ê2A2,H0=1μvê3×E0,

the directions of E0 and H0 being as depicted in Fig. 1.2, with θ given by

(1.60c)tanθ=A2A1.

More generally, one can consider a superposition of the two linearly polarized waves (Eqs. 1.59a and 1.59b; which we have referred to as the x-polarized wave and the y-polarized wave, respectively) but now with a phase difference, say, δ:

(1.61)E=E1+eiδE2,H=H1+eiδH2.

If we consider the y-polarized wave in isolation, the multiplication of E2, H2 with the phase factor eiδ does not change the nature of the wave, since only the common phase of oscillations of the electric and magnetic intensities is changed. But the above superposition (Eq. 1.61) with an arbitrarily chosen value of the phase angle δ (which we assume to be different from 0 or π; see later) does imply a change in the nature of the resulting wave in that, while the instantaneous electric and magnetic field intensities and the propagation vector still form a right-handed triad, the electric and the magnetic intensities now no longer point along fixed directions as in the case of a linearly polarized wave.

Thus, for instance, if one chooses A1 = A2(= A), say, and δ=π2or−π2, then the tip of the directed line segment representing the instantaneous electric field intensity E (which here denotes the real electric intensity vector rather than its complex representation) describes a circle in the x-y plane of radius A, while a similar statement applies to H as well. For δ=π2, the direction of rotation of the vector is counterclockwise (ie, from the x-axis toward the y-axis), while the rotation is clockwise for δ=−π2(check this out; see Fig. 1.3). These are said to correspond to left-handed and right-handed circularly polarized waves, respectively .

In the quantum theoretic description, photons with states of polarization corresponding to δ=π2and δ=−π2are commonly referred to as right-handed and left-handed photons respectively.

As seen above, a superposition of the x-polarized wave and the y-polarized wave with the phase difference δ = 0 results in a linearly polarized wave with the direction of polarization (ie, the line of oscillation of the electric intensity at any given point in space; in Fig. 1.2 we take this point to be at the origin of a chosen right-handed coordinate system) inclined at an angle θ given by Eq. (1.60c). The value δ = π, on the other hand, again gives a linearly polarized wave, with θ now given by tanθ=−A2A1(check this statement out).

For the general case in which δ is different from the special values 0, π (and for A1 = A2, the values δ=±π2), one finds that the tip of the electric intensity vector describes an ellipse in the x-y plane (where, for concreteness, we consider the variation of the electric intensity at the origin of a chosen right-handed coordinate system). Once again the direction of rotation of the electric intensity vector can be counterclockwise or clockwise, depending on the value of δ, corresponding to left-handed and right-handed elliptic polarization, respectively (see Fig. 1.4) .

3.2.2 Standard Basis Vectors

Continuing on with the preliminaries, let us next consider Fig. 3.4. This figure shows a three-dimensional space with the vector a′ = (1, 2, 2) appearing as a directed line segment. To set up this coordinate space, we define a special set of zero-one coordinate vectors, denoted ei′, as

vector e1′ of unit length in the positive (by convention) x direction:

e1′=(1,0,0)

vector e2′ of unit length in the positive y direction:

e2′=(0,1,0)

vector e3′ of unit length in the positive z direction:

e3′=(0,0,1)

We shall continue to let 0′, the zero vector, denote the origin of the space. As suggested in the discussion of vector addition and scalar multiplication of a vector in Chapter 2, we can now write the vector a′ = (1, 2, 2) as a linear combination of the coordinate vectors:

1e1′+2e2′+3e3′=1(1,0,0)+2(0,1,0)+2(0,0,1)=(1,0,0)+(0,2,0)+(0,0,2)                        a′=(1,2,2)

Note that what we have done is to perform scalar multiplication followed by vector addition, relative to the coordinate vectors ei′. We shall call the ei′ vectors a standard basis and comment later on the meaning of basis vectors, generally.

Note, further, that if we had the oppositely directed vector –a′, this could also be represented in terms of the standard basis vectors as the linear combination:

−1e1′−2e2′−2e3′=(−1,0,0)+(0,−2,0)+(0,0,−2)=(−1,−2,−2)

In this case –a′ would extend in the negative directions of x, y, and z.

What is shown above in particularized form can be generalized in accordance with the discussion of linear combinations of vectors in Chapter 2. As recalled:

Given p n-component vectors b1′, b2′, …, bp′ the n-component vector

a′=∑i=1pkibi′=k1b1′+k2b2′+⋯+kpbp′

is a linear combination of p vectors, b1′, b2′ …, bp′ for any set of scalars ki (i = 1, 2, …, p).

In the illustration above we have p = 3 basis vectors, each containing n = 3 components. The components of the vector a′ = (1, 2, 2) involve p = 3 scalars. The bi′ vectors in the more general expression above correspond to the specific ei′ vectors in the preceding numerical illustration.

The introduction of a set of standard basis vectors allows us to write any n-component vector a′, relative to a standard basis of n-component ei′ vectors, as

a′=∑i=1naiei′

where ai denotes the ith component of a′, and each of the n basis vectors has a 1 appearing in the ith position and zeros elsewhere. In this special case of a linear combination, the number of vectors p equals the number of components in a′, namely, n.

Figure 3.5, incidentally, shows a′ in terms of the triple of numbers (1,2, 2). This point representation, as we now know, is equally acceptable for representing a′ since the vector is already positioned with its tail at the origin.

The important point to remember is that a′, itself, can be represented as a linear combination of other vectors–in this case, the standard basis vectors ei′. In a sense the ei′ represent a standard scale unit across the three axes. Any projection of a′, then, can be considered as involving (signed) multiples of the appropriate ei′ vector.

With these preliminaries out of the way, we can now introduce the central concept of the chapter, namely, the Euclidean space and the associated idea of the distance between two points, or vector termini, in Euclidean space. This idea, in turn, leads to the concepts of angle and length.

Definition 1

A vector is a 1 × n or n × 1 matrix.

A1 × n matrix is called a row vector while an n × 1 matrix is a column vector. The elements are called the components of the vector while the number of components in the vector, in this case n, is its dimension. Thus,

[123]

is an example of a 3-dimensional column vector, while

[t2t−t0]

is an example of a 4-dimensional row vector.

The reader who is already familiar with vectors will notice that we have not defined vectors as directed line segments. We have done this intentionally, first because in more than three dimensions this geometric interpretation loses its significance, and second, because in the general mathematical framework, vectors are not directed line segments. However, the idea of representing a finite dimensional vector by its components and hence as a matrix is one that is acceptable to the scientist, engineer, and mathematician. Also, as a bonus, since a vector is nothing more than a special matrix, we have already defined scalar multiplication, vector addition, and vector equality.

A vector y (vectors will be designated by boldface lowercase letters) has associated with it a nonnegative number called its magnitude or length designated by ‖y‖.

1.7 The geometry of vectors

Vector arithmetic can be described geometrically for two- and three-dimensional vectors. For simplicity, we consider two dimensions here; the extension to three-dimensional vectors is straightforward. For convenience, we restrict our examples to row vectors, but note that all constructions are equally valid for column vectors.

A two-dimensional vector v = [a b] is identified with the point (a, b) on the plane, measured from the origin a units along the horizontal axis and then b units parallel to the vertical axis. We can then draw an arrow beginning at the origin and ending at the point (a, b). This arrow or directed line segment, as shown in Fig. 1.4, represents the vector geometrically. It follows immediately from Pythagoras's theorem and Definition 2 of Section 1.6 that the length of the directed line segment is the magnitude of the vector. The angle associated with a vector, denoted by θ in Fig. 1.4, is the angle from the positive horizontal axis to the directed line segment measured in the counterclockwise direction.

Solution The vectors are drawn in Fig. 1.5. Using Pythagoras's theorem and elementary trigonometry, we have, for v,

‖v‖=(2)2+(4)2=4.47,tanθ=42=2,andθ=63.4∘.

For u, similar computations yield

‖u‖=(−1)2+(1)2=1.14,tanθ=1−1=−1,andθ=135∘.

To construct the sum of two vectors u + v geometrically, graph u normally, translate v so that its initial point coincides with the terminal point of u, being careful to preserve both the magnitude and direction of v, and then draw an arrow from the origin to the terminal point of v after translation. This arrow geometrically represents the sum u + v. The process is depicted in Fig. 1.6 for the two vectors defined in Example 1.

To construct the difference of two vectors u − v geometrically, graph both u and v normally and construct an arrow from the terminal point of v to the terminal point of u. This arrow geometrically represents the difference u − v. The process is depicted in Fig. 1.7 for the two vectors defined in Example 1. To measure the magnitude and direction of u − v, translate it so that its initial point is at the origin, being careful to preserve both its magnitude and direction, and then measure the translated vector.

Both geometrical sums and differences involve translations of vectors. This suggests that a vector is not altered by translating it to another position in the plane providing both its magnitude and direction are preserved.

Many physical phenomena such as velocity and force are completely described by their magnitudes and directions. For example, a velocity of 60 miles per hour in the northwest direction is a complete description of that velocity, and it is independent of where that velocity occurs. This independence is the rationale behind translating vectors geometrically. Geometrically, vectors having the same magnitude and direction are called equivalent and they are regarded as being equal even though they may be located at different positions in the plane.

A scalar multiplication ku is defined geometrically to be a vector having length ‖k‖ times the length of u with direction equal to u when k is positive, and opposite to u when k is negative. Effectively, ku is an elongation of u by a factor of ‖k‖ when ‖k‖ is greater than unity, or a contraction of u by a factor of ‖k‖ when ‖k‖ is less than unity, followed by no rotation when k is positive, or a rotation of 180 degrees when k is negative.

Example 2

Find −2u and 12vgeometrically for the vectors defined in Example 1.

Solution To construct −2u, we double the length of u and then rotate the resulting vector by 180°. To construct 12v, we halve the length of v and effect no rotation. These constructions are illustrated in Fig. 1.8. ■

Figure 1.8.

Basic Ideas

Our line equation is based on vectors (Figure 5.1). Recall that we draw a vector as a directed line segment (using an arrow) stretching from a point called the tail to a point called the head. Two such segments are considered to represent the same vector if they have the same direction and the same length. They need not have the same tail. When we look at a directed line segment we are actually looking at just one representative of a vector, although this distinction often gets lost when we refer to the directed line segment loosely as a vector.

If a vector has its tail at A = (xA, yA) and its head at B = (xB,yB), then the vector from A to B can be represented with numerical components as 〈xBs − xA, yB − yA〉. If A and B are three-dimensional points, then we would have 〈xB – xA, yB – yA, zB – zA 〉. Vectors will be denoted by lowercase boldface letters, such as a, p, u, x; for example, p = 〈2, 3, 4〉. Boldfacing is awkward for work on a chalkboard or on paper, so you might p→use in place of p, for example.

Vectors can be added, subtracted, and multiplied by scalars. For example, if u = 〈1, 5, −3〉 and v = 〈−6, 2, −1〉, then u + v = 〈−5, 7, −4〉 and u − v = 〈7, 3, −2,〉 4u = 〈4, 20, −12〉 and −2u = 〈−2, −10, 6〉.

We rely here on a variety of facts about vectors that are usually introduced in the study of multivariable calculus. Three of these facts concern the geometric meaning of equality, addition, and scalar multiplication of vectors.

1.

If u is the vector from B to A and v is the vector from D to C, then u and v have the same components if and only if the directed segment from B to A has the same length and direction as the directed segment from D to C.

If u and v are vectors, we can find u + v as follows: Select a directed line segment representing u and then place a representative of v so its tail is at the head of u. The vector from the tail of u to the head of v is u + v.

If u is a vector and r a scalar (an ordinary number), then (a) if r > 0, ru points in the same direction as u and its length is r times as long, and (b) if r < 0, ru points in the opposite direction to u but it is |r| times as long.

The vector from the origin to a point P is called the position vector for P, and we denote it p. In general, we denote a position vector by the lowercase boldface version of the capital letter used for the point. If P has coordinates (xP, yP, zP), then the position vector p = 〈xP, yP, zP〉. As you can see, there is very little difference between points and position vectors, so you might wonder why we don't use the same notation, P, to stand for both or why we introduce vectors at all. One reason is that a vector can be repositioned to have its tail at any point we please. This is important in establishing the theory of vector calculations and in some applications. However, in our work, to avoid confusion, often visualize a position vector as having its tail at the origin.

Suppose we are given the coordinates of two points A and B and a third point X is given and we want to know if it is on the line determined by A and B. The points could be in two-dimensional space or three-dimensional space — our solution method will be the same. For definiteness, let's say we are in three dimensions, so A = (xA, yA, zA) and B = (xB, yB, zB), X = (x, y, z). X is on line AB↔(Figure 5.1) if and only if vector x − a points in the same or opposite direction as b − a. This means

(5.1)x-a=t(b-a)for some scalar t.

(In Figure 5.1, t = 1.5.) Equation (5.1) yields

(5.2)x=a+t(b-a).

This can be thought of as a set of equations, one for each coordinate of the space we are in. For example, in three dimensions we have

(5.3)x=xA+t(xB−xA),y=yA+t(yB−yA),z=zA+t(zB−zA).

In two-dimensional space we have just the first two of these equations.

Equation (5.2) is perhaps the easiest way to remember the parametric equations, but when we use them we often need (Eq. 5.3) or the two-dimensional version of Eq. (5.3).