Vector Addition in .NET framework

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5.3 Vector Addition
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The resultant or vector sum of a number of vectors, all in the same plane, is that vector in the plane which would produce the same effect as that produced by all the original vectors acting together.
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CHAPTER 5 Practical Applications
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If two vectors and have the same direction, their resultant is a vector R whose magnitude is equal to the sum of the magnitudes of the two vectors and whose direction is that of the two vectors. See Fig. 5.6(a). If two vectors have opposite directions, their resultant is a vector R whose magnitude is the difference (greater magnitude smaller magnitude) of the magnitudes of the two vectors and whose direction is that of the vector of greater magnitude. See Fig. 5.6 (b).
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Fig. 5.6
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In all other cases, the magnitude and direction of the resultant of two vectors is obtained by either of the following two methods. (1) Parallelogram Method. Place the tail ends of both vectors at any point O in their plane and complete the parallelogram having these vectors as adjacent sides. The directed diagonal issuing from O is the resultant or vector sum of the two given vectors. Thus, in Fig. 5.7(b), the vector R is the resultant of the vectors and of Fig. 5.7(a). (2) Triangle Method. Choose one of the vectors and label its tail end O. Place the tail end of the other vector at the arrow end of the first. The resultant is then the line segment closing the triangle and directed from O. Thus, in Figs. 5.7(c) and 5.7(d), R is the resultant of the vectors and .
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Fig. 5.7 EXAMPLE 5.4 The resultant R of the two vectors of Example 5.2 represents the speed and direction in which the boat travels. Figure 5.8(a) illustrates the parallelogram method; Fig. 5.8(b) and (c) illustrate the triangle method.
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The magnitude of R
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13mi/h rounded. 0.3333 and 18 .
From Fig. 5.8(a) or (b), tan u
Thus, the boat moves downstream in a line making an angle 18 with the direction in which it is headed or making an angle 90 72 with the bank of the river. (See Sec. 4.7 for rounding procedures.)
Fig. 5.8
CHAPTER 5 Practical Applications
5.4 Components of a Vector
The component of a vector along a line L is the perpendicular projection of the vector very useful to resolve a vector into two components along a pair of perpendicular lines. on L. It is often
EXAMPLE 5.5 In Fig. 5.8(a), (b), and (c) the components of R are (1) 4 mi/h in the direction of the current and (2) 12 mi/h in the direction perpendicular to the current. EXAMPLE 5.6 In Fig. 5.9, the force F has horizontal component Fh 30 . Note that F is the vector sum or resultant of Fh and Fv.
F cos 30 and vertical component Fp
F sin
Fig. 5.9
5.5 Air Navigation
The heading of an airplane is the direction (determined from a compass reading) in which the airplane is pointed. The heading is measured clockwise from the north and expressed in degrees and minutes. The airspeed (determined from a reading of the airspeed indicator) is the speed of the airplane in still air. The course (or track) of an airplane is the direction in which it moves relative to the ground. The course is measured clockwise from the north. The groundspeed is the speed of the airplane relative to the ground. The drift angle (or wind-correction angle) is the difference (positive) between the heading and the course.
Fig. 5.10
In Fig. 5.10: ON is the true north line through O / NOA is the heading OA the airspeed AN is the true north line through A / NAW is the wind angle, measured clockwise from the north line AB the windspeed / NOB is the course OB the groundspeed / AOB is the drift angle
CHAPTER 5 Practical Applications
Note that there are three vectors involved: OA representing the airspeed and heading, AB representing the direction and speed of the wind, and OB representing the groundspeed and course. The groundspeed vector is the resultant of the airspeed vector and the wind vector.
EXAMPLE 5.7 Figure 5.11 illustrates an airplane flying at 240 mi/h on a heading of 60 when the wind is 30 mi/h from 330 . In constructing the figure, put in the airspeed vector at O, then follow through (note the directions of the arrows) with the wind vector, and close the triangle. Note further that the groundspeed vector does not follow through from the wind vector. In the resulting triangle: Groundspeed tan Course 30/240 60 2(240)2 0.1250 and 67 10 (30)2 242 mi/h
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