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In the gures, the vector quantities all represent distances measured o at di erent angles relative to some speci c reference line. " In the gures, let us suppose that the distance represented by A happens to lie exactly " is at an angle of zero degrees with respect to along the reference line; that is, suppose that A the reference line; symbolically this can be shown by writing that " " A jAj=08 " Or, if A were measured (for example) at an angle of, say, 208 with respect to the reference line, this would be shown by writing " " A jAj=208 " " In Fig. 2-A, B is a distance, of magnitude jBj, measured from point P at an angle let us " denote by b degrees relative to the vector A, as shown in the gure. (We re assuming " is at 08 relative to the reference line.) The resultant of these two measurements is a that A " distance of magnitude jRj at, let us say, an angle of r degrees relative to the reference line. Algebraically, the whole operation can be expressed by writing that " " " jAj=08 jBj=b8 jRj=r8 " " " " or, in a more abbreviated form, A B R, which says that vector R is equal to the sum of " and B. " vectors A " " " " Now, in regard to actually nding the value of R, where R A B, two procedures, one graphical and the other mathematical, are available. Here we ll mainly emphasize the graphical procedure, as follows. At the beginning of this discussion we de ned that quantities are truly vector quantities only if they obey the PARALLELOGRAM LAW OF ADDITION. This simply means " " " " that, vectorially speaking, in order for R to be the true vector sum of A and B, R must be " and B as opposite " equal to the DIAGONAL OF THE PARALLELOGRAM having A sides of the parallelogram. With this in mind, consider Fig. 2-A; we know, from actual experience, that displace" " ment A, plus displacement B, produces exactly the same nal result as would the single ". But study of Figs. 3-A and 4-A shows that R is equal to the diagonal of the " vector R " and B as opposite sides; thus displacement is a true vector " parallelogram having A quantity. " " To illustrate the above graphical procedure, consider Fig. 5-A, in which A and B are given to be two vector quantities at an angle a, as shown. Let the problem be to nd, " " " " graphically, the resultant vector R, where R A B. " and B are given to be vector quantities, the rst step is to construct a paralle" Since A " " logram with A and B as opposite sides, as in Fig. 6-A. The diagonal of the parallelogram, " " " " drawn from the junction of A and B, is the resultant of the two vectors A and B; that is, it " " " A B. is the graphical solution of the equation R
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In the foregoing discussion we ve referred to displacement as the basic vector quantity. A number of other quantities, such as force and velocity, are also vector quantities. Of special interest to us, however, is the fact that rms values of alternating currents can be treated as vector quantities. If we have THREE vectors, we rst nd the resultant of any two, then combine that resultant with the third vector to get the nal resultant. We proceed in the same way to nd the sum of any number of vectors. In this way we can state that the rule for nding the sum of n vectors by geometric means is as follows. Keeping the directions of the vectors unchanged, move them by translation (that is, without rotation) until the tail of the second touches the head of the rst, the tail of the third touches the head of the second, and so on, for all the n vectors. " The sum or resultant of the n vectors is the vector R, which is drawn from the tail of the rst vector to the head of the last nth vector. " " " " The above rule is illustrated for the addition of four vector quantities, A; B; C; D, in Fig. 7-A. This gure is thus the geometric solution to the vector equation " " " " " A B C D R, where the vectors are given to have the magnitudes and directions as shown.
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