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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 twodimensional vectors and by V3 the set of all threedimensional vectors. More generally, we will later need to consider the set Vn of all
ndimensional vectors. An ndimensional vector is an ordered ntuple:
 Vectors in n dimensions are used to list various quantities in an organized way. For instance,
the components of a sixdimensional vector
p1 , p2 , p3 , p4 , p5 , p6 p might represent the prices of six different ingredients required to make a particular product.
Fourdimensional 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 5E13(pp 838847) 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.
 Fall '09
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