Vectors [Essentials of Vector and Phase Analysis]

The basic idea of vectors has already been introduced in the two previous SECTIONs. Stated simply, a vector is a line that represents quantity and direction. This direction may be com pared with a known reference, or with some other vector. It can be said that vectors are directed lines because the line points in a specific direction. The term directed-line segment is also used, assuming that the line itself extends indefinitely, but in the problem at hand only a small part (segment) is being used.

Vectors can be compared with scalars, also called scalar quantities, which are the quantities with which we work most often. Such quantities may be a temperature, a number of specific items, area or volume of a figure, population, attendance at an event, or many other similar examples. These are all specific amounts of what is being measured, but they do not involve direction. To say that the temperature is 72° implies only a magnitude, no sense of direction. Temperature is stated completely by specifying the number of degrees, assuming the measure of a degree is known. So it can be said that a scalar quantity has only magnitude ; it does not imply any direction of action.

Fig. 3-1. Four different vectors, one in each quadrant.

Other quantities with which we deal can be expressed in terms of magnitude, but are more meaningful if direction is also stated. These include statements of velocity, number of pounds of push or pull, or other cases in which action is per formed in a specific direction. Technically, speed is considered to be a scalar quantity; velocity is vectorial and includes both speed and direction. Although not explicitly directional, the vector idea is used in certain aspects of electrical circuits, for example the phases of voltages and currents. This was shown briefly in the latter part of the preceding SECTION and will be used to a much greater extent in the last two SECTIONs.

Fig. 3-1 includes four different vectors, one in each of the four quadrants. The length of each line is proportional to the magnitude of the quantity being represented. The arrow of each vector points in the direction in which the represented quantity is acting, and in every case the vectors originate at the origin (0). That end of the vector at which the arrow is located can be called the terminal, or point, end. The opposite end can be called the initial, or starting, end (or point). As long as all the vectors in Fig. 3-1 begin at the origin, they could be labeled OA, OB, OC, and OD. It is evident that these are vectors representing vector quantities. But without the diagram the designations OA, OB, etc. may be confusing. So to indicate that they are vectors - a, a, etc. could be used, and in some cases only the line is used above the letter as a, OA, OB - , etc. The first letter normally indicates the initial point; the second letter is the terminal point. OA, for example, means that the vector lies in the direction of A. Other forms of notation are also used, and one of the most popular is a single letter, such as A, B, etc. Later, you will see that R, XL, Z, and other electronics abbreviations are used as vector symbols. And in some texts boldface type is used such as A, B, etc. In addition to expressing a vector in terms of letter designations, they may also be expressed in ways that not only

designate them as vectors, but which also give their magnitude and angular direction. One of these is A/ θ, which indicates a vector length of A acting in the direction of angle θ. This is known as the polar form of a vector because this length radiates from the origin which could be called a pole. Along this same line, circular paper calibrated in degrees of angle is called polar graph paper. Or, as will be explained later, a vector can be expressed in terms of rectangular co-ordinates including an X and a Y dimension. Other symbols may be used for vectors, but regardless of what symbol is used, the writing should make it clear that vectors are being represented and what the symbols indicate. In this writing vector quantities are considered almost entirely so there should be no confusion along that line.

In performing mathematical operations with two or more vectors, they are not treated the same as scalar quantities.

Scalars are simply numbers and are handled as any other numbers in legitimate mathematical operations. Vectors, how ever, cannot be treated so simply. Not only must the magnitudes be considered, but the directions must also be included in order to obtain correct results. Much of this SECTION and all of the next SECTION is devoted to mathematical operations involving vectors. This is necessary because in order to use vectors in AC circuitry we must be able to add, subtract, multiply, and divide by using vector notation. The word resultant will occur many times, and it will be used to indicate the single total vector that is obtained from a specific combination of individual vectors.

At this point in the text it may be well to introduce several other terms related to vectors, because they may be encountered in various texts on electrical and electronic principles. First, it is reasonably standard to assume that two forces pushing or pulling an object can be represented by vectors. Such a condition is shown in Fig. 3-2A in which Fl is a force of 8 pounds attempting to pull an object in a certain direction. F2 is a force of 10 pounds that is pulling in another direction as shown. The resultant force (also a vector) would be at an angle between F1 and F2, and would have a specific number of pounds of pull.

A vector diagram could also be three-dimensional, for example one vector pulling to the right, one to the left, and one either up or down. Those shown here are plane vectors; they are not three-dimensional. And in this writing only plane vectors are considered.

(A) Pulling forces (B) Voltage and current relationship.

Fig 3-2 Vectors representing relationships of forces.

(A) Phasor.

Fig. 3-3 (B) Vector.

Phasor and vector diagrams.

Fig. 3-2B shows specific types of vectors which occur very often in electrical and electronic work. In this case they represent the voltage and current relationship in an electrical circuit.

They may be referred to as phasors (rather than vectors) be cause they represent phase relationships of the circuit. Phasors can be defined as quantities representing sine waves in amplitude and phase position by length and direction from a given reference line. Comparing this diagram with Fig. 3-2A we find that there are differences in the quantities represented. It is difficult to say that voltage and current are acting in a specific direction; but we can say that if voltage is at some instantaneous point in its sine-wave cycle, the current is a specific number of degrees behind the voltage in reaching that same angle of rotation. Thus, a phasor is used instead of a vector.

With these ideas in mind, you can see that Fig. 3-3A would be a phasor diagram and Fig. 3-3B would be a vector diagram.

We have to agree that there are definite relationships between the oppositions in a circuit and the current-voltage relation ship in that same circuit. So the use of two terms here (phasor and vector) could create confusion. Still another definition states that phasors are any complex-number quantities associated with AC circuit theory. Using this definition, Fig. 3-3B would also be a phasor diagram as we will conclude after studying complex numbers in the next SECTION.

The term sinor is also used to some extent and applies to sine-wave positions varying with respect to time. It could be said that the sinor is a special form of phasor, in fact one definition states that "phasors that specify sinusoidal time functions are termed sinors." As an example, a sine wave represents a voltage or current varying at a specific rate and plotted against time. It could be said that a phasor diagram shows two or more separate sinors, but this statement is not a clear-cut definition.

In order to simplify this text, the term vector is used exclusively for several reasons:

1. The term vector is readily understood, and its use extends back many years, even for electrical quantities.

2. Use of the single term prevents confusion, especially when there seems to be no sharp dividing line between the various terms.

3. Actually, a phasor is a specific form of vector, using the latter in a general sense.

4. Regardless of what is being represented by the directed lines, the methods of working with them are the same.

As long as it is known what is being represented by the directed lines, these differences in terminology should present no real problem, hence the use of the single term vector.

VECTOR COMPONENTS

Any vector can be described (or defined) in several different ways. Symbols for vectors were introduced earlier, also the polar form of notation, but there is still another way which lends itself quite well to trigonometric solutions. For example, consider the vector in Fig. 3-4; it starts at the origin and terminates at point (8, 4). Plotted on the co-ordinate axes, as shown, the vector can be expressed in terms of its X and Y values, called rectangular components. The process of determining the horizontal and vertical components from the vector is referred to as resolving the vector, or resolution of the vector.

Fig. 3-4. A vector expressed by rectangular components.

If a triangle is completed by drawing a line from the end of the vector to the X axis (shown as dashed line), we find that the side opposite angle θ is 4 units in length. The adjacent side is 8 units in length. So originating at the origin the rectangular components are (8,4). Expressing the vector in this form is referred to as the rectangular form of vector notation because the measurements are with respect to the rectangular co ordinates. The point numberings being used were introduced along with rectangular co-ordinates in the preceding SECTION.

Depending on what is given, either form can be converted to the other by trigonometric calculations. In Fig. 3-4 the rectangular form is given, and in this form the rectangular co ordinates give the lengths of the two short sides of a right triangle. The actual vector length is the hypotenuse of that same triangle and it forms the angle θ with the adjacent side.

Using the symbols given, the vector length can be determined by substituting into the Pythagorean formula:

Z2 ,- X2 + y2 Z --= VX2 + Y2 Z - v82 + 42 = V80 = 8.94 units Y

Angle θ may be determined by 0 = arc tan - X =arc tan 0.5 = = 26.6°.

So the actual vector, in polar form, is 8.94 /26.6°. Both Z and 0 are stated in terms of the closest digits rather than by large number of decimal places. This practice is also followed in succeeding sections of the text because methods are more important than extreme accuracy when learning the basic processes.

In practical calculations either form may be given, and it may be necessary to convert from one form to the other. When the rectangular form is given, the polar form is determined by: V X2 ± y2 / arc t an 1.. using the lettering of Fig. 3-4.

When the polar form is given, the horizontal component is Z times cos O; the vertical component is Z times sin 0, again using the lettering of Fig. 3-4. Bear in mind, however, that these two forms and the calculations involved apply only to a single vector and are based on the relationships existing in a right tri angle. As we shall see in the next section of this SECTION, when working with more than one vector, the angle between them must be considered, and this is not necessarily a right angle. However, each individual vector can be resolved and this is one of the methods of solution which are introduced.

As shown in the preceding SECTION, unless otherwise specified, angle 0 is assumed to be in standard position, that is, measured with respect to 0°. Normally 0° is on the horizontal line and to the right of the origin. The 0° line may be called the initial side of the vector and the actual vector the terminal side, especially when considering rotating vectors as in AC. These terms were also introduced previously.

VECTOR ADDITION

As previously indicated, vectors can be combined in various ways, just as other types of math expressions can be, as long as the angles are considered along with the magnitudes. Scalar quantities can be added directly, for example:

15 tubes + 10 tubes = 25 tubes 3 books + 2 books = 5 books

But unless vectors are at the same angle, the magnitudes can not be added directly. The fact that the quantities are acting in different directions prevents that. Also remember that only vectors representing like quantities can be added. For example, to combine voltage and current into a resultant quantity has no meaning. Two voltages, however, can be combined to give a resultant voltage; this idea applies similarly to other vector quantities.

Fig. 3-5. Parallel vectors.

(A) Acting in same direction.

(B) Acting in opposite directions.

Addition of Two Vectors If two vectors are parallel (at the same angle), they are acting in the same direction so that their magnitudes can be considered like scalar quantities and added directly. The resultant is still a vector acting at the same angle as the two separate quantities. As an example, suppose that two forces, 20 pounds and 15 pounds, are acting at an angle of 45° with respect to some reference. The resultant is 35 pounds acting at the same angle, 45°. They can be plotted on the same vector line as shown in Fig. 3-5A, and parallel vectors in any direction are handled similarly. A vector would be considered equal to an other vector only when both have the same magnitudes and are acting at the same angle.

Vector lines may be parallel with respect to each other, but acting in exactly opposite directions. There is then a 180° angular difference between them, and they are not considered to be parallel vectors. The effects of such vectors are cancellable since they directly oppose each other. In Fig. 3-5B a force of 20 pounds is acting at an angle of 45° and a force of 15 pounds is acting at an angle of 225°. The resultant vector is 5 pounds (20 - 15) acting at an angle of 45°. The magnitudes are sub tractive, and the resultant vector is in the direction of that vector having the larger magnitude. If both vectors had exactly the same magnitude the resultant force would be zero.

Bear in mind that the individual angles need not be 45° and 225°. Vectors are subtractive whenever they directly oppose each other, at any angles.

It may seem that the 0° and 180° angular differences would not necessarily be useful, but that is not the case. In electrical and electronic circuits these angles occur quite often. For ex ample, two inductive reactances act at the same angle in a circuit and are directly additive. But inductive and capacitive reactances add in opposition to each other and are subtractive.

Also, in amplifier circuits, signals may be in phase or 180° out of phase with each other. So the vectors representing them would be either additive or subtractive. These subjects, how ever, are covered in greater detail in a subsequent SECTION.

Now consider a situation in which the vectors are not at 0° nor 180°, as in Fig. 3-6. Here F1 is a force of 10 pounds acting on an object at the origin, and acting at an angle of 80°. F2 is a force of 15 pounds acting on the same point but at an angle of 30°. The resultant is the single force acting at a specific angle that would act exactly like the two single forces applied at the two separate angles. The resultant here is at some angle be tween 30° and 80° and is labeled R.

Fig. 3-6. Adding two vectors that are not at 0 ° or

Fig. 3-7. Another example of adding vectors that are not at 0° or 180°.

Fig. 3-8. Adding vectors by placing them end to end.

The resultant in this case, including force and angle, has been determined by drawing a parallelogram with the two given forces as adjacent sides. A parallelogram is a four sided figure in which opposite sides are parallel and equal in length, and in which opposite angles are equal to each other.

Line F1' is the same length and at the same angle as line F1.

Similarly line F2' is the same length and at the same angle as line F2. The angle in each case would be measured with respect to where the dashed line (extended) would cross the X axis.

The resultant vector is the diagonal of the parallelogram drawn from the point of origin of the two original vectors. Its length can be measured to determine the number of pounds of effective force, and the angle can be measured (with a protractor) to determine the angle at which the resultant force is acting. From Fig. 3-6 we can conclude that the resultant is a force of 22.5 pounds acting at an angle of 50°. In drawing the resultant vector be careful to draw only that diagonal that starts at the point of origination of the two separate vectors. This is necessary because each parallelogram has two possible diagonal lines. We can draw certain definite conclusions regarding such an arrangement. When the vectors are at the same angle, the sum is greater than either of the two separate vectors. At 180° the difference vector is shorter than the longer of the two separate vectors. For angles between 0° and 180° the resultant is be tween the two separate angles, and the resultant vector length depends on the angular difference. As the two individual angles come closer to each other, the resultant length increases, reaching its maximum when the angles are equal. Actual resultant length is also determined by the length of the individual vectors, becoming longer as the separate vectors are made longer.

If the forces represented by F1 and F2 are the same, then the resultant angle would be exactly half way between the two separate angles. However, if they are not equal, the resultant is closer to that vector having the larger magnitude. In Fig. 3-6 F2 represents a larger force than Fl so that the resultant is closer to 30° than to 80°. If F2 were increased in length (F1 remaining the same), the resultant angle would be even closer to 30°. Any convenient scale can be used to determine the lengths of the lines, and each inch of vector length represents a specific number of pounds of force. Generally, the larger the drawing, the more accurate will be the results obtained by this graphic method of solution. When the drawing is too small, even the width of the pencil line may decrease the accuracy.

Another similar example is that shown in Fig. 3-7. In this case, F1 is a force of 15 pounds at 110°, and F2 is a force of 25 pounds at 50°. The resultant is a force of 35 pounds acting at 72°. Notice that in every example of this type the results are only as accurate as the graphical presentation, and they may not always be exact. This is a definite disadvantage of any graphical means of solution, but in the presentation of the subject they are invaluable for a visual description of the pro cess involved. A similar but different method of solution is that of Fig. 3-8, which is a representation of the same problem solved in Fig. 3-6. Instead of starting both vectors at the origin, only one is started there. Then the second begins at the terminal point of the first. The dashed line completing the triangle is the resultant and it is the same, both in magnitude and direction, as the diagonal of the parallelogram of Fig. 3-6.

It does not matter which vector starts at the origin; the same resultant is obtained either way. But they must be arranged so that the initial point of the second vector joins the terminal point of the first. Then the resultant is the vector that connects the initial point of the first vector to the terminal point of the second. This is sometimes called the head-to-tail method, and it is especially useful when more than two vectors are involved.

It involves a smaller number of lines, and therefore does not appear as complicated as the parallelogram method. This method is also graphic, however, and subject to the same inaccuracies as other graphic methods.

Fig. 3-9. Adding two vectors that are at right angles.

Fig. 3-10. Using rectangular components for two vectors in the first quadrant.

Fig. 3-11. Using rectangular components for one vector in the first quadrant and the other in the second.

If two forces are at right angles to each other, the problem is simplified because trig can be used to give a more exact resultant. One angle of the triangle is then a right angle. Fig.

3-9 shows an example in which the parallelogram method of solution has been used. The head-to-tail method could also have been used, in which case the Y vector would appear as the right side of the parallelogram. In either method of solution the resultant magnitude can be measured with a scale and a pro tractor. When the two vectors to be added are at 90° (Fig. 3-9), the length of the diagonal is the same regardless of which diagonal is used. To avoid confusion with respect to the resultant angle, the diagonal which begins at the origin should be used.

As long as the right triangle does exist in this case, a trig solution can be made as follows. The Pythagorean relationship is used to determine the length of the resultant:

R -= V X2 ± rj = V 4 2 + 3 2 = V 25 = 5 0 can be determined by: 3 0 = arc tan - 4 = 36.9°

In this problem the resultant is 5 units acting at an angle of 36.9°. Either arc sine or arc cosine could also be used once the length of the resultant has been determined.

Another possibility for using rectangular components is the solution of Fig. 3-10. F1 and F2 are vectors whose resultant solution has been determined by the parallelogram method.

But these vectors were drawn so that the X and Y components would be whole numbers to better illustrate another method of solution. Fl terminates at point (3,9) ; F2 terminates at point (13,6). The resultant vector terminates at point (16,15). If the X values (3 and 13) are added, the Y values (9 and 6) are added. The X result is 16; the Y result is 15. This provides an algebraic method of vector addition, first the X components are added, and then the Y components are added. The rectangular components of the resultant vector are the sums of the rectangular components of the individual vectors.

Fig. 3-11 shows the same idea applied to two other vectors, one in the first quadrant, the other in the second. F1 terminates at (-4, 12), F2 terminates at (8,7). Adding the X components (-4 and 8) gives 4, and adding the Y components (12 and 7) gives 19. The resultant terminates at point (4, 19), and a vector presentation such as that shown gives the same result.

This resolution method is especially important in electrical calculations because of the 90° phase shift of current and voltage in reactive components.

When a vector is given in polar form, it can be converted to rectangular co-ordinates by:

X = length x cos Y = length x sin θ

There is no algebraic method of adding polar-form vectors unless their angles are the same, for example: 3/20° + 2/20° = 5/20° Otherwise, the rectangular co-ordinates must be added, and then, if necessary, the resultant converted back to polar form.

More use will be made of these ideas in later SECTIONs with reference to complex numbers. These are numbers which use specific notations to denote vectors, either in polar or rectangular form.

Three methods of vector addition have been shown. These are the parallelogram, head-to-tail, and resolution methods of which the last is preferred because it is an algebraic method, does not require drawing utensils, and probably gives the best accuracy. There is, however, a trigonometric method that can be used to solve for the resultant when the individual vectors are not at right angles to each other. This involves the study of oblique triangles and the Law of Cosines. But because it is used to such a limited degree in vectors associated with electronics, that method is not included here. Volume 2 of Electronics Math Simplified contains a section on oblique triangles and the use of the Law of Cosines.

Addition of Three Vectors The basic ideas involved in adding more than two vectors are the same as those shown for the two-vector additions, except that more operations are involved. In this section the addition of three vectors is considered, but the ideas can be applied to even more. Most practical problems do not include more than three so that is the limitation here. Any of the methods previously shown can be used for three vectors-the algebraic method is usually the one preferred.

In Fig. 3-12 the parallelogram method has been used to add three vectors which start from the origin. The process used is to find the resultant of any two, then to add that resultant to the third vector and thus obtain a resultant for the entire combination. F1 is 6 units in length at 100°, F2 is 8 units at 60°, and F3 is 10 units at -20°. In Fig. 3-12, F1 and F2 were added and the diagonal of the parallelogram is the resultant (R1). Then F3 and R1 were added by constructing a second parallelogram, the diagonal of which is RT the total resultant. As measured, the resultant is approximately 15.5 units in length at an angle of about 37.5°.

Fig. 3-12. Parallelogram method of adding three vectors which start from the origin.

Fig. 3-13. Using the head-to-tail method for adding three vectors.

Plotting any other combination would give the same result.

As an example, F2 and F3 could be added and then the result ant combined with Fl to obtain the total resultant vector. F 1 and F3 could be added and the resultant added to F2.

The head-to-tail method has been used on the same problem in Fig. 3-13. F1 was started at the origin, F2 started at the terminal end of F1, and then F3 started at the terminal end of F2. The R1 resultant (same as in Fig. 3-12) has been drawn, but it is not necessary in the solution of the problem. RT is again the total resultant and is drawn from the origin to the terminal end of F3. This is sometimes referred to as closing the polygon. When three vectors are added head-to-tail, the resultant is that vector required to close the four-sided figure.

The figure constructed, however, is not normally a parallelogram. From Fig. 3-13, resultant RT is about the same as previously found, 15.5 units at 37.5°. The same problem can also be solved by the resolution method of obtaining the X and Y components of the vector.

Using the same scale as in Figs. 3-12 and 3-13, the X and Y components are approximately as shown here: F1 X = -1, Y = 6 F2 X = 4, Y = 7 F3 X = 9.3, Y = -3.5 RT X = 12.3, Y = 9.5

Adding the X's and the Y's separately produces X = 12.3 and Y = 9.5 for the total resultant vector. The length of RT can be determined by the Pythagorean relationship.

RT = V- X2 + Y = V 12.32 + 9.52 = V 151.29 + 90.25 = V 241.54 = 15.54

This result is very close to that obtained by graphic means.

Y 9.5 θ = arc tan -- = arc tan X 12.3 = arc tan .7724 = 37.7° (approx.)

This too is about the same as determined by the graphic method. But let's consider even another possibility. If each of the polar-form vectors were converted to rectangular form by algebraic means and then combined by addition, results which are more accurate would be obtained. As we shall see later, most practical vector problems are solved by algebraic means.

VECTOR SUBTRACTION

In algebra a number is subtracted by changing its sign and then performing algebraic addition. This is also true for subtracting vectors. When the sign of a vector is changed, it is the 3/25° + 2/25° = 5/25° same as changing the angle by 180°. When adding, When subtracting, 3/25° - 2/25° -= 1/25° Effectively, the direction of the second vector was reversed, causing it to directly oppose the action of the first. So the second expression could also be shown as:

3/25° + 2/205° = 1/25°

The parallelogram method can be used in the subtraction of vectors as well as in addition. Two vectors are added by determining the diagonal of the parallelogram of which the two original vectors are adjacent sides. Referring back to Fig. 3-6, F1 and F2 were added to give the resultant (R), as shown: F1 + F2 = R

Considering the same problem values let's assume a sub traction such that: R – F1 = F2

Fig. 3-14. Using the parallelogram method to subtract vectors.

Fig. 315. Subtracting the resultant vector (R) from vector F1.

Fig. 3-16. Subtraction method by changing the sign of the vector being subtracted.

As shown in Fig. 3-14, R is 22.5/50°, and F1 is 10/80°. In this method of subtraction a parallelogram is set up such that the vector being subtracted from (the minuend) is the diagonal. As a first step, the ends of the vectors are connected (dashed lines have been used). Then the parallelogram has been completed as shown. Measuring F2 gives 15/30°. Comparing this with Fig. 3-6, it agrees, and the diagram is the same parallelogram as obtained in the addition problem. So effectively we have determined the vector that would combine with F1 (addition) to give the resultant R. This shows that sub traction in vectors is the inverse of addition, just as it is in any other form of mathematics.

Notice that it makes a difference as to what is being subtracted. The vector being subtracted from is the diagonal in every case. For example, in Fig. 3-14 if vector F2 had been subtracted from R (11 - F2), the diagram would have been the same as shown. F2 would have been given in the original problem and F1 would have been the difference of the subtraction process.

If the process had been altered to F1 - R the difference would not be the F2 value found previously because Fl would be the diagonal. The other value would be that as shown by F2 in Fig. 3-15. This new vector is approximately 14/210°. Another subtraction method which can be used is based on the changing of signs mentioned at the beginning of this section on subtraction. It is possibly much simpler than the parallelogram method because there is less chance of performing the wrong subtraction. Referring back to Fig. 3-15 we find that 10/80° was subtracted from 22.5/50. Then after changing the sign of the vector being subtracted the problem becomes: 22.5/50° - 10/80° The graphic representation of this problem is shown in Fig. 3-16. Changing F1 by 180° causes the problem to read: 22.5/50° + 10/260°

This is true because -10/80° is the same vector as +10/260° shown as Fr. By completing the parallelogram and drawing the diagonal, F2 is found to be 15/30°, which agrees with the previous solution. When using this method, change only the sign of the vector being subtracted. Any other change produces erroneous results.

Fig. 3-17. Subtracting X and Y components algebraically.

The X and Y components can be subtracted algebraically in much the same way as used in vector addition. The only difference is the changing of signs of the vector being subtracted.

An example is the problem shown in Fig. 3-17. Here F1 has rectangular components of X = -4, and Y= 11. F2 values are X = 9 and Y = 14. If F2 is being subtracted from F1, the signs of the F2 components are changed and they become X = -9 and Y = -14. Adding them to the Fl values gives X = -13 and Y = -3. In Fig. 3-17 F2 has been shifted by 180° as de noted by F2', and the diagonal of the parallelogram is labeled R. Notice that it terminates at point (-13, -3).

One word of caution before leaving subtraction. In most algebraic (and vectorial) expressions subtraction is indicated in the problem itself. For example, 8/15° - 3/15° shows sub traction; do not change any of the signs. But if the problem is to subtract 3/15° from 8/15° then the sign of the 3/15° must be changed. The problem then reads 8/15° - 3/15°, as previously indicated.

MULTIPLICATION AND DIVISION

Multiplication and division of vectors do not have as clear a meaning in graphical form as do addition and subtraction. For example, the addition of two vectors to determine a resultant vector is a logical process. We can see its meaning. Multiplication and division are also logical processes, but not as clear as to actual meaning. Suppose that vector F1 (3/20°) is multi plied by F2 (4/45'). The resultant product is: (3/_2_0°) (4/45°) = 12/65° In the process the magnitudes are multiplied together, and then the angles are added. These vectors are shown in Fig. 3-18. With addition and subtraction, only like quantities can be combined. For example, volts and amperes cannot be added or subtracted because no meaningful unit would result, even though the vectors themselves could be combined. With multiplication and division, however, different units can be combined as long as the same scale is used for both basic units. This does not mean that just any combination can be multiplied together because there is not a corresponding unit for every combination. But for example, Ohm's law states that current multiplied by resistance equals circuit voltage. So current (in amperes) times resistance (in ohms) equals circuit voltage (in volts). This is an example of a product combination that is meaningful with respect to the units involved. However, if we were graphing such a product it would be necessary to let the same length of line represent an ampere, an ohm, and a volt. Otherwise the magnitude of the product would be incorrect.

In the example of Fig. 3-18 only the basic units of the terms are considered, there is no statement of what the units may be.

The two original vectors are 3/20° and 4/45°; suppose that each unit on the graph is Y4 inch in length. Fl would be 3 / 4 inch long on the graph, and F2 would be 1 inch long. R is 12/65° so its length would be 3 inches on the 65° line. As you can see, the resultant was obtained by algebraic, not graphical, means.

Fig. 3-18. Finding the resultant by considering only the basic units of the terms.

However, there is one interesting relationship that can be used even though it is not used to any extent in practical work.

If a vector of unit length (y4 inch in our example) is laid out on the 0° line, an angle 0 is formed by connecting the ends of the unit line and Fl. As shown, the same angle (0) exists when a line is drawn between the ends of F2 and R. As is probably obvious, multiplication and division are best performed by algebraic means, either in rectangular or polar form. All of these processes are described and illustrated more thoroughly in the next two SECTIONs. But for now we can assume that in polar-form multiplication the magnitudes are multiplied together and the angles added. This means that: (F1/01) (F2/0g) = F1F2/01 + 02

In division the magnitudes are divided and the angles subtracted, as follows:

F1101 F1

F2/02

In some cases multiplication (or division) may be used as part of an entire problem, but it has no real meaning in itself.

One example can be found in the product-over-the-sum method for determining total impedance:

Z1Z2 ZT = Z1 + Z2

Multiplying Z1Z2 gives a specific product, but it has no specific meaning unless it is combined with the rest of the equation.

Then Zr has definite meaning. In electronics problems concerning vectors a number of such examples exist.

ROTATING VECTORS

The basic idea underlying a rotating vector was described previously, but in this SECTION it is carried further. In SECTION 1 a simple generator was used to generate a sine wave and the amplitude at any instant of time was shown to be dependent on the angle at which the magnetic lines of force were cut. The same idea can be illustrated vectorially, using the diagram of Fig. 3-19. The radius of the circle is assumed to be a rotating vector, and several different instantaneous positions are shown.

Positive (CCW) rotation is assumed, beginning at zero degrees.

Fig. 3-19. Representing a rotating vector and several different instantaneous positions.

The length of the vector represents the maximum amplitude of the vector and is constant for any given problem, even though the instantaneous amplitude is maximum at only two angles in the complete sine-wave cycle. At any instantaneous position of rotation a vertical line drawn from the end of the rotating vector to the horizontal axis represents the instantaneous amplitude. The length of this line is proportional to the sine of that angle, because with a given length of vector sin θ = 0/H. The vector is the hypotenuse so that this proportionality holds true.

Several examples are illustrated in Fig. 3-19. At the 30° angle the dashed line is one-half the length of the vector (hypotenuse). At 60°, the dashed line is .866 of the vector length; since both are above the horizontal axis, they are positive. A second-quadrant angle of 135° is also shown, and sin 135° = .707. The instantaneous amplitude at that angle is

.707 of the maximum amplitude. At 225° the sine is -.707; therefore the instantaneous amplitude is also negative, as it is throughout the third and fourth quadrants. This also verifies the statement made previously, that at any angle, the instantaneous amplitude of the sine wave is equal to the maximum amplitude multiplied by the sine wave of the angle.

(A) Sine waves. (B) Vectors.

Fig. 3-20. Two ways of showing lead and lag.

When two sine waves are involved, they may or may not be in phase as previously indicated. Fig. 3-20 shows two ways of showing lead or lag. In Fig. 3-20A both sine waves are shown, and by proper labeling it can be indicated whether current is leading or lagging. The entire cycles are shown, because to show only small sections of cycles may prove to be a hazard in determining phase relationships. However, the diagram in Fig. 3-20B, with proper labeling, can show the same thing with two vectors, although some texts call this a phasor diagram, as previously indicated. The vectors can be shown at any angles in the cycles, but the angular difference between them should always be the same. By careful checking it is seen that the angular differences in both diagrams are the same.

In later SECTIONs it will be pointed out that it is more or less standard to show one vector at 0° and the other at some other angle. This is easier, both for drawing and for reference purposes. The same type of diagram can represent phase relation- ships of two voltages, two currents, or of a voltage and a cur rent. All of the diagrams would be the same except for labeling.

More than two vectors can be included in either type of dia gram, as will be seen in the last two SECTIONs.

QUIZ

1. A vector is 12 units in length at 35°. What are the X and Y rectangular co-ordinates?

2. The terminal end of a vector lies at point (6,4). Express the vector in polar form.

3. By any convenient method add the vectors 5/200 and 6/58°.

4. By any convenient method add the vectors 8/ 7 35° and 6/40°.

5. By any convenient method add the vectors which have terminal ends at points (-5,6) and (8,3).

6. Subtract vector 7/18° from 15/72°.

7. Subtract vector (4,6) from (-8,5).

8. Multiply vectors 6/15° and 3/200.

9. Multiply vectors 8/-38° and 5/32°.

10. Divide vector 16/48° by 8/18°.