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Elementary mathematical and computational tools for electrical engineers using Matlab - Chapter 7

Vectors Một vector 2-D được xác định bởi chiều dài của nó và góc nó làm cho một trục tham chiếu (thường là trục x). Vector này là đại diện đồ họa bởi một mũi tên. Các đuôi của mũi tên được gọi là điểm ban đầu của vector và đầu mũi tên là điểm thiết bị đầu cuối. Hai vectơ bằng nhau khi cả hai chiều dài và góc với một trục tham chiếu bằng nhau. | 7 Vectors 7.1 Vectors in Two Dimensions 2-D A vector in 2-D is defined by its length and the angle it makes with a reference axis usually the x-axis . This vector is represented graphically by an arrow. The tail of the arrow is called the initial point of the vector and the tip of the arrow is the terminal point. Two vectors are equal when both their length and angle with a reference axis are equal. 7.1.1 Addition The sum of two vectors u v w is a vector constructed graphically as follows. At the tip of the first vector draw a vector equal to the second vector such that its tail coincides with the tip of the first vector. The resultant vector has as its tail that of the first vector and as its tip the tip of the just-drawn second vector the Parallelogram Rule see Figure 7.1 . The negative of a vector is that vector whose tip and tail have been exchanged from those of the vector. This leads to the conclusion that the difference of two vectors is the other diagonal in the parallelogram Figure 7.2 . 7.1.2 Multiplication of a Vector by a Real Number If we multiply a vector fl by a real number k the result is a vector whose length is k times the length of v and whose direction is that of v if k is positive and opposite if k is negative. 7.1.3 Cartesian Representation It is most convenient for a vector to be described by its projections on the x-axis and on the y-axis respectively these are denoted by v1 v2 or vx vy . In this representation 2001 by CRC Press LLC 3 FIGURE 7.1 Sum of two vectors. FIGURE 7.2 Difference of two vectors. u u1 u2 u1 ê1 u2 ê2 7.1 where ê1 and ê2 are the unit vectors length is 1 parallel to the x-axis and y-axis respectively. In terms of this representation we can write the zero vector the sum of two vectors and the multiplication of a vector by a real number as follows 2001 by CRC Press LLC 0 0 0 0ê1 0ê2 7.2 u v w u1 v1 u2 v2 u1 v1 ê1 u2 v2 ê2 7.3 ku ku1 ku2 ku1 ê1 ku2 ê2 7.4 Preparatory Exercise Pb. 7.1 Using the above definitions and .