Fowles01

Fowles01 - CHAPTER 1 FUNDAMENTAL CONCEPTS VECTORS 1.1 ^ j(a A B =(i ^ ^ k = i 2 ^ k j ^ ^ j ^ A B =(1 4 1 = 6 1 2 ^ ^ j ^ j(b 3 A 2 B = 3(i ^ 2 ^ k

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CHAPTER 1 FUNDAMENTAL CONCEPTS: VECTORS 1.1 (a) AB ˆˆ ˆ ˆ ˆ () ( ) 2 i j jk i jk +=+++=+ + K K 1 2 (1 4 1) 6 +=++ = K K (b) 32 ˆ 3 ( ) 2 3 2 AB ij j k ij −=+−+=+ K K k (c) (1)(0) (1)(1) (0)(1) 1 ⋅= + + = K K (d) ˆ ˆ ˆ 110 ( 10 ) ( 01 ) 011 ijk i j k i j k ×= = −+ −+ − =−+ K K 1 2 1 3 ×=++ = K K KK 1.2 (a) ˆ ˆˆ ˆ ˆ 2 4 (2)(1) (1)(4) 6 ABC i j i ⋅+ = +⋅++= + + = K ˆ ˆ 3 4 (3)(0) 4 i jk j +⋅= ++⋅= + + = K (b) 210 101 8 040 ⋅× = = K () () 8 ABC ABC ×⋅= ⋅×= (c) ( ) ( ) ( ) ˆ ˆ 42 4 4 8 A CB A B C ik j ×× =⋅ −⋅ = +− =−+ K K K 4 ( ) ( ) ˆ ˆ 02 4 4 4 AB C C AB CBA CAB ij i k ××= −××= − ⋅  =− + + = +  K K K K K K K K 1
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1.3 2 2 22 () () ( 2) ( 2) ( 0 ) ( 3) 5 cos 51 4 4 AB a a a a a a AB a aa θ ⋅+ + == = K K 1 5 cos 53 14 =≈ ° 1.4 (a) ˆ ˆˆ Ai jk = ++ K : body diagonal K K ˆˆ ˆˆ 3 AA A i ij jk k =⋅= ⋅+ ⋅= (b) B = + K : face diagonal K 2 BB B =⋅= K (c) ˆ 111 110 ijk B =×= KK K CA (d) 11 0 32 AB ⋅− == = K K cos 90 = D 1.5 sin BB ACA C ==×= K K K sin y B CC A cos AC u = cos x u A 2 xy A uB A B C A A BA ××  =+  ×  K B A K K K K 1 u =+× A K K K 1.6 23 ˆ ˆ d d d it jt kt i j t k dt dt dt dt 2 t dA α βγ β + = + + K γ 2 2 ˆ ˆ 26 dA j dt K 2
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1.7 () () () ( ) ( )( ) 2 1 2 AB q q q q q =⋅= + −+ = − + K K 03 3 2 0 , q 21 qq −− = 1 = or 2 1.8 2 22 2 AB AB A B +=+ ⋅+=++⋅ KK K K K K 2 2 A B += + + K K   Since , cos A B AB AB θ ⋅= K K AB A B + ≤+ K K K K cos cos = K K cos B B 1.9 Show ( ) C A CB A B C ×× =⋅ −⋅ K K K or ( ˆ ˆˆ xyz x xy yz z x z ijk B B A C A B A BC CCC ×= + + + + K ˆ xx yxy zxz xxx yyx zzx C ABC ABC i =++−−− ˆ x yyy zyz xxy zzy C j +++−−− ˆ x yzy zzz xxz yyz C k ) x xz ˆ ˆ ˆ xyx xzx j ABC k +−− =+ + ++−− 3
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ˆ ˆˆ xy yz zy zx xz yx ijk AAA z B CB C −−− ( ) () ˆ ˆ ˆ yxy yyx zzx zxz zyz zzy xxy xyx xzx xxz yyz yzy i ABC ABC j ABC kABC −−+ =+ + +− + 1.10 yA sin θ = 1 2 xy y B x xy yB xy AB 2s i n  = + −=+ −=   Α A Α= B × K K 1.11 ˆ xyz ijkAAA C AB B B B B B CCC CCC ⋅× = = KK K K z z z AB BBB BAC CCC = −= K × 1.12 x z x G C G A G G Let = (Ax,Ay,Az), A K B K = (0,By,0) and C G = (0,Cy,Cz) K C z is the perpendicular distance between the plane A , B K and its opposite.
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This note was uploaded on 03/09/2008 for the course PHYS 301 taught by Professor Mokhtari during the Fall '04 term at UCLA.

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Fowles01 - CHAPTER 1 FUNDAMENTAL CONCEPTS VECTORS 1.1 ^ j(a A B =(i ^ ^ k = i 2 ^ k j ^ ^ j ^ A B =(1 4 1 = 6 1 2 ^ ^ j ^ j(b 3 A 2 B = 3(i ^ 2 ^ k

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