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Unformatted text preview: S E C T I O N 13.2 Vectors in Three Dimensions (ET Section 12.2) 313 To show that→ AO =→ AO we must express u in terms of v , w and z . We have v + w z u = ⇒ u = v + w z Substituting into (2) we get→ AO = 1 2 v + 1 4 w + 1 4 ( v + w z ) = 1 2 v + 1 4 w + 1 4 v + 1 4 w 1 4 z = 3 4 v + 1 2 w 1 4 z (3) By (1) and (3) we conclude that→ AO =→ AO . It means that the points O and O are the same point, in other words, the segment F H and E G bisect each other. 66. Prove that two vectors v = a , b and w = c , d are perpendicular if and only if ac + bd = 0. SOLUTION Suppose that the vectors v and w make angles θ 1 and θ 2 , which are not π 2 or 3 π 2 , respectively, with the positive xaxis. Then their components satisfy a = v cos θ 1 b = v sin θ 1 ⇒ b a = sin θ 1 cos θ 1 = tan θ 1 c = w cos θ 2 d = w sin θ 2 ⇒ d c = sin θ 2 cos θ 2 = tan θ 2 y x v w q 1 q 2 That is, the vectors v and w are on the lines with slopes b a and d c , respectively. The lines are perpendicular if and only if their slopes satisfy b a · d c = 1 ⇒ bd = ac ⇒ ac + bd = We now consider the case where one of the vectors, say v , is perpendicular to the xaxis. In this case a = 0, and the vectors are perpendicular if and only if w is parallel to the xaxis, that is, d = 0. So ac + bd = · c + b · = 0. 13.2 Vectors in Three Dimensions (ET Section 12.2) Preliminary Questions 1. What is the terminal point of the vector v = 3 , 2 , 1 based at the point P = ( 1 , 1 , 1 ) ? SOLUTION We denote the terminal point by Q = ( a , b , c ) . Then by the definition of components of a vector, we have 3 , 2 , 1 = a 1 , b 1 , c 1 Equivalent vectors have equal components respectively, thus, 3 = a 1 a = 4 2 = b 1 ⇒ b = 3 1 = c 1 c = 2 The terminal point of v is thus Q = ( 4 , 3 , 2 ) . 2. What are the components of the vector v = 3 , 2 , 1 based at the point P = ( 1 , 1 , 1 ) ? SOLUTION The component of v = 3 , 2 , 1 are 3 , 2 , 1 regardless of the base point. The component of v and the base point P = ( 1 , 1 , 1 ) determine the head Q = ( a , b , c ) of the vector, as found in the previous exercise. 3. If v = 3 w , then (choose the correct answer): (a) v and w are parallel. (b) v and w point in the same direction. SOLUTION The vectors v and w lie on parallel lines, hence these vectors are parallel. Since v is a scalar multiple of w by a negative scalar, v and w point in opposite directions. Thus, (a) is correct and (b) is not. 314 C H A P T E R 13 VECTOR GEOMETRY (ET CHAPTER 12) 4. Which of the following is a direction vector for the line through P = ( 3 , 2 , 1 ) and Q = ( 1 , 1 , 1 ) ? (a) 3 , 2 , 1 (b) 1 , 1 , 1 (c) 2 , 1 , SOLUTION Any vector that is parallel to the vector→ P Q is a direction vector for the line through P and Q . We compute the vector→ P Q :→ P Q = 1 3 , 1 2 , 1 1 = 2 , 1 , ....
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This note was uploaded on 07/15/2009 for the course MATH 210 taught by Professor Hubscher during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 Hubscher
 Vectors

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