Chapter 13

# Similarly in two dimensions we dene i 1 0 and j 0 1

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Unformatted text preview: directions of the positive x-, y-, and z-axes. Similarly, in two dimensions we deﬁne i 1, 0 and j 0, 1 . (See Figure 17.) y z j k (0, 1) 0 x i i (1, 0) F IGURE 17 Standard basis vectors in V™ and V£ y x (a) If a (b) a 1, a 2 , a 3 , then we can write a a 1, a 2 , a 3 a a™ j a¡ i 0 a 2 a 1, 0, 0 0, a 2 , 0 a 1 1, 0, 0 y (a¡, a™) j a 2 0, 1, 0 a 3 0, 0, 1 a1 i a2 j 0, 0, a 3 a3 k Thus, any vector in V3 can be expressed in terms of the standard basis vectors i , j, and k. For instance, x 1, 2, 6 i 2j 6k (a) a=a¡ i+a™ j Similarly, in two dimensions, we can write z a 3 a1, a2 a1 i a2 j (a¡, a™, a£) See Figure 18 for the geometric interpretation of Equations 3 and 2 and compare with Figure 17. a a£ k a¡ i y x a™ j (b) a=a¡ i+a™ j+a£ k EXAMPLE 5 If a of i , j, and k. i 2j 3 k and b 4i 7 k, express the vector 2 a 3 b in terms SOLUTION Using Properties 1, 2, 5, 6, and 7 of vectors, we have FIGURE 18 2a 3b 2i 2j 3k 2i 4j 3 4i 6k 12 i 7k 21 k 14 i 4j 15 k A unit vector is a vector whose length is 1. For instance, i , j, and k are all unit vectors. In general, if a 0, then the unit vector that has the same direction as a is u 4 In order to verify this, we let c the same direction as a. Also u 1 a a 1 a . Then u ca c a a a c a and c is a positive scalar, so u has 1 a a 1 5E-13(pp 838-847) 840 ❙❙❙❙ 1/18/06 11:14 AM Page 840 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 6 Find the unit vector in the direction of the vector 2 i j 2 k. SOLUTION The given vector has length 2i j s2 2 2k 1 2 2 2 s9 3 so, by Equation 4, the unit vector with the same direction is 1 3 2i j 2 3 2k 1 3 i 2 3 j k Applications Vectors are useful in many aspects of physics and engineering. In Chapter 14 we will see how they describe the velocity and acceleration of objects moving in space. Here we look at forces. A force is represented by a vector because it has both a magnitude (measured in pounds or newtons) and a direction. If several forces are acting on an object, the resultant force experienced by the object is the vector...
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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.

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