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Fig. 7-A
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" " " To geometrically nd the di erence of two vectors, A B, we draw the vector B in " accordance with Fig. 1-A, then combine it with vector A by means of the parallelogram law in the usual way. This is illustrated in Fig. 8-A, which shows the geometric solution to " " " the vector equation R A B.
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It is sometimes convenient to represent a given vector as being the sum of two or more " component vectors. This is illustrated in Fig. 9-A, in which a given vector A is repre" " 0 and A 00 (A prime and A double sented as being the vector sum of the two components A prime).
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It s especially useful, when nding the sum of a number of vectors, to express each vector in terms of its HORIZONTAL AND VERTICAL COMPONENTS. To do this, we place the tails of all the vectors at the origin O of the x, y plane, then resolve each vector into horizontal components, all lying on the x-axis, and vertical components, all " " lying on the y-axis. This is illustrated in Fig. 10-A for two given vectors A and B, with their horizontal and vertical components denoted by the subscripts h and v respectively.
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" The above procedure applies to the problem of nding the resultant sum, R, of any number of vectors. The advantage of the procedure is that the total HORIZONTAL " COMPONENT of R is the simple algebraic sum of all the horizontal components, and " the total VERTICAL COMPONENT of R is the simple algebraic sum of all the vertical components.
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Increment (Delta) Notation
The symbol D is the Greek letter delta. A term such as Dq denotes an optionally SMALL CHANGE in the value of a variable q, and is read as delta q. Note that Dq does not mean delta times q.
Appendix
Terms such as Dq and Dt are called increments of the variables q and t, and denote small changes in the values of q and t. As used here, q denotes a total amount of electric charge in coulombs, and t denotes a total amount of time in seconds, counted from some optionally chosen time at which t 0. Hence the ratio of the increments, Dq=Dt, is the AVERAGE time rate of change of q in coulombs per second, which is given the name amperes. At a time t Dt the total charge is q Dq; thus, as Dt becomes smaller and smaller, Dq also becomes smaller and smaller, and the ratio Dq=Dt comes closer and closer to being the EXACT value of current, i, owing at the beginning of the interval of time Dt, at time t. This idea is expressed mathematically by writing that
Dt!0
Dq i exact current at a time t Dt
which says that the limiting value of the ratio Dq=Dt, as Dt is allowed to approach zero as a limit, is the exact value of current at a time t. If the mathematical relationship between q and time t is known, then, using the formulas of di erential calculus, the value of current i at any time t can be calculated. Mathematically, the above limit is denoted by the symbol dq/dt, which is read as dee q, dee t. Thus dq coulombs per second amperes dt
Note 6.
Similar Triangles. Proof of Eq. (98)
Two triangles are called similar if their ANGLES are all equal. Thus the two right triangles in Fig. 11-A are similar triangles. By de nition, if two triangles are similar their corresponding angles are equal (in Fig. 11-A the corresponding angles are 30 and 30, 60 and 60, and 90 and 90 degrees). The corresponding sides are A and A 0 , B and B 0 , and C and C 0 , as shown. Note that the ratio A=B is equal to the ratio A 0 =B 0 , the ratio A=C is equal to the ratio A 0 =C 0 , and so on. In Fig. 11-A the two similar triangles are in a position such that their corresponding sides are PARALLEL; thus, if the corresponding sides of two triangles can be shown to be mutually PARALLEL, this establishes that the two are similar triangles. Also, if two triangles are similar they can always be moved and rotated into a position such that their corresponding sides are PERPENDICULAR, as illustrated in Fig. 12-A. Thus, if the corresponding sides of two triangles can be shown to be mutually PERPENDICULAR, this is su cient to establish that the two are similar triangles.
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