4 0 3 4 2 0 2 1 5 5 2 1 5 1 3 3 5 6 1 2 6 3 15 3b

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Unformatted text preview: 4 2 a a b 02 4, 0, 3 4 a b s25 5 2, 1, 5 2, 0 1, 3 4, 0, 3 5 2, 1, 8 2, 1, 5 4 2 ,0 3b 3 5b 2a 32 2 4, 0, 3 1, 3 2, 1, 5 3 5 5 6, 1, 2 2 ,3 1 ,3 5 6, 3, 15 2, 1, 5 8, 0, 6 10, 5, 25 2, 5, 31 We denote by V2 the set of all two-dimensional vectors and by V3 the set of all threedimensional vectors. More generally, we will later need to consider the set Vn of all n-dimensional vectors. An n-dimensional vector is an ordered n-tuple: |||| Vectors in n dimensions are used to list various quantities in an organized way. For instance, the components of a six-dimensional vector p1 , p2 , p3 , p4 , p5 , p6 p might represent the prices of six different ingredients required to make a particular product. Four-dimensional vectors x, y, z, t are used in relativity theory, where the first three components specify a position in space and the fourth represents time. a a1, a 2, . . . , a n where a1, a 2, . . . , a n are real numbers that are called the components of a. Addition and scalar multiplication are defined in terms of components just as for the cases n 2 and n 3. Properties of Vectors If a, b, and c are vectors in Vn and c and d are scalars, then 1. a b b 3. a 0 a 5. c a 7. cd a 2. a a b 4. a b ca c a ca 8. 1a c da da c 0 6. c cb a b a da These eight properties of vectors can be readily verified either geometrically or algebraically. For instance, Property 1 can be seen from Figure 4 (it’s equivalent to the Parallelogram Law) or as follows for the case n 2: a b a 1, a 2 b1 Q c b (a+b)+c =a+(b+c) b a+b b+c P FIGURE 16 b1, b2 a 1, b2 a2 a1 b1, a 2 b1, b2 b2 a 1, a 2 a We can see why Property 2 (the associative law) is true by looking at Figure 16 and l applying the Triangle Law several times: The vector PQ is obtained either by first constructing a b and then adding c or by adding a to the vector b c. Three vectors in V3 play a special role. Let a i 1, 0, 0 j 0, 1, 0 k 0, 0, 1 5E-13(pp 838-847) 1/18/06 11:13 AM Page 839 SECTION 13.2 VECTORS ❙❙❙❙ 839 Then i , j, and k are vectors that have length 1 and point in the...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.

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