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Unformatted text preview: Problern 3.4 Given A = *2 i3 + 21 and B = tB' +y2 + 2Bz: (a) find B' and B, if A is Parallel to B; (b) find a rel4tion between B' and B, If Lis perpendicular to B' Solution: (a) If A is parallel to B, then their directions are equal or opposite: it : lin, or From theycomPonent, A/lAl : +B/lBl, ?293 +2 , *,Bx+92+28, _ _ _  frT   JT+4TE . l a )  L m:@ which can only be solved for the minus sign (which means that A and B must point in opposite directions for fhem to be parallel). Solving fot B! + Bl, / 2 ,'\ 2 20 ni+ ni:  \/r4  4: ; . \  J / 98, ^ 2\66 4 : : I :  v/i6D) "r 3Jt4 3 From the )ccomponent, 2 J14 and, from the zcomPonent, , Ar:A' This is consistent with our result for ni + A? These results could also have been obtained by assuming 04 was 0o or l80o and solving lAllBl : fA'B, or by solving A x B = 0' (rii if a is perpendicular to B, then their dot product is zero (see Section 31.4). Using Eq. (3.21), 0 : A ' B : 2 B r  6 + B z , or B ,   6  2 8 , . There are an infinite number of vectors which could be B and be perpendicular to A, but their x and zcomponents must satisff this relation' This result could have also been obtained by assuming 0AB :90" and calculating lAllBl :lAxBl. Problem 3.6 Given vectors A : i29* 23 andB : t3 tZ,finda vector C whose magnitude is 9 and whose direction is perpendicular to both A and B. Solution: The cross product of two vectors produces a new vector which is perpendicular to both of the original vectors. Two vectors exist which have a magnitude of 9 and are orthogonal to both A and B: one which is 9 units l6ng in the direction of the unit vector parallel to A x B, and one in the opposite direction. c: +effi: +e (*29+23) x (i3 22) l(f,2i+el; x(?322)l : +gW= *(f;1.34 +y8.67 +zz.o). ' JF*1V*9 Pr:oblern 3.10 Find an expression for the unit vector directed toward the point P located on the zaxis at a height lr above the xy plane from an arbitrary point Q: (*,y,5) in the plane z: 5. Solution: PointP is at (0,0,ft). VectorA from Q: (r,y,3) toP: (0,0,&) is: A : i(0 .r) +i(0 v) +2(h+ s) : *r  vv+2(h+ 5), lAl : ["2 +y2 +(h+stz1tlz, ^ A f.x 9y+2(h+s) *  lAl  lxz lYz + (h+ s)2ltl2' Problem 3.27 Asectionof asphereis describedby0 Problem 3....
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 Spring '08
 RAKOV
 Electromagnet

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