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Chapter 7. Vectors

Chapter 7. Vectors - 7 Vectors 7.1 Vectors in Two...

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© 2001 by CRC Press LLC 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 is a vector constructed graphically as fol- lows. 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 dif- ference 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 by a real number k , the result is a vector whose length is k times the length of , and whose direction is that of if k is pos- itive, 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 ( v 1 , v 2 ) or ( v x , v y ). In this representation: r r r u v w + = r v r v r v

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© 2001 by CRC Press LLC (7.1) where ê 1 and ê 2 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 vec- tor, the sum of two vectors, and the multiplication of a vector by a real num- ber as follows: FIGURE 7.1 Sum of two vectors. FIGURE 7.2 Difference of two vectors. r u u u u ê u ê = = + ( , ) ( ) ( ) 1 2 1 1 2 2
© 2001 by CRC Press LLC (7.2) (7.3) (7.4) Preparatory Exercise Pb. 7.1 Using the above definitions and properties, prove the following identities: The norm of a vector is the length of this vector. Using the Pythagorean the- orem, its square is: (7.5) and therefore the unit vector in the direction, denoted by ê u , is given by: (7.6) All of the above can be generalized to 3-D, or for that matter to n -dimensions. For example: (7.7) r 0 0 0 0 0 1 2 = = + ( , ) ê ê r r r u v w u v u v u v ê u v ê + = = + + = + + + ( , ) ( ) ( ) 1 1 2 2 1 1 1 2 2 2 ku ku ku ku ê ku ê r = = + ( , ) ( ) ( ) 1 2 1 1 2 2 r r r r r r r r r r r r r r r r r r r r r r r r r r r u v v u u v w u v w u u u u u k lu kl u k u v ku kv k l u ku lu + = + + + = + + + = + = + − = = + = + + = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 r u u u 2 1 2 2 2 = + r u ê u u u u u = + 1 1 2 2 2 1 2 ( , ) ê u u u u u u u n n = + +… 1 1 2 2 2 2 1 2 ( , , , )

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© 2001 by CRC Press LLC 7.1.4 MATLAB Representation of the Above Results MATLAB distinguishes between two kinds of vectors: the column vector and the row vector. As long as the components of the vectors are all real, the dif- ference between the two is in the structure of the array. In the column vector case, the array representation is vertical and in the row vector case, the array representation is horizontal. This distinction is made for the purpose of
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Chapter 7. Vectors - 7 Vectors 7.1 Vectors in Two...

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