Fowles01 - CHAPTER 1 FUNDAMENTAL CONCEPTS VECTORS 1.1 ^ j(a...

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CHAPTER 1 FUNDAMENTAL CONCEPTS: VECTORS 1.1 (a) A B ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) 2 i j j k i j k + = + + + = + + K K 1 2 (1 4 1) 6 A B + = + + = K K (b) 3 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3( ) 2( ) 3 2 A B i j j k i j = + + = + K K k (c) (1)(0) (1)(1) (0)(1) 1 A B = + + = K K (d) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 1 0 (1 0) (0 1) (1 0) 0 1 1 i j k A B i j k i j k × = = + + = + K K 1 2 (1 1 1) 3 A B × = + + = K K K K 1.2 (a) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ 2 4 (2)(1) (1)(4) (0)(1) 6 A B C i j i j k + = + + + = + + = K ( ) ( ) ˆ ˆ ˆ ˆ 3 4 (3)(0) (1)(4) (1)(0) 4 A B C i j k j + = + + = + + = K K K (b) ( ) 2 1 0 1 0 1 8 0 4 0 A B C × = = − K K K ( ) ( ) 8 A B C A B C × = × = − K K K K K K (c) ( ) ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ ˆ 4 2 4 4 8 A B C A C B A B C i k j i j k × × = = + = + K K K K K K K K K 4 ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ ˆ 0 2 4 4 4 A B C C A B C B A C A B i j i k i k × × = − × × = − = − + + = + K K K K K K K K K K K K 1

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1.3 2 2 2 2 ( )( ) (2 )(2 ) (0)(3 ) 5 cos 5 14 5 14 A B a a a a a a AB a a a θ + + = = = K K 1 5 cos 53 14 θ = ° 1.4 (a) ˆ ˆ ˆ A i j k = + + K : body diagonal K K ˆ ˆ ˆ ˆ ˆ ˆ 3 A A A i i j j k k = = + + = (b) ˆ ˆ B i j = + K : face diagonal K 2 B B B = = K (c) ˆ ˆ ˆ 1 1 1 1 1 0 i j k B = × = K K K C A (d) 1 1 0 3 2 A B AB θ = = = K K cos 90 θ = D 1.5 sin B B A C AC θ = = × = K K K sin y B C C A θ = = cos A C AC u θ = = K K cos x u C C A θ = = 2 x y A B A u B A B C C C A A A A B A × × = + = + × K K K B A K K K K K K 2 2 1 u A B A A = + × A K K K 1.6 ( ) ( ) ( ) 2 3 ˆ ˆ ˆ ˆ ˆ ˆ 2 3 d d d i t j t k t i j t k dt dt dt dt 2 t dA α β γ α β + = + + K γ = + 2 2 ˆ ˆ 2 6 d A j k t dt β γ = + K 2
1.7 ( )( ) ( )( ) ( )( ) 2 1 2 A B q q q q q = = + + = + K K 0 3 3 2 0 ( ) , q ( ) 2 1 q q = 1 = or 2 1.8 ( ) ( ) 2 2 2 2 A B A B A B A B + = + + = + + K K K K K K A B K K 2 2 2 2 A B A B AB + = + + K K Since , cos A B AB AB θ = K K A B A B + + K K K K K K cos cos A B AB A B A B θ θ = = K K K K cos B B θ 1.9 Show ( ) ( ) ( ) C A C B A B C × × = K K K K K K K A B K K or ( ) ( ˆ ˆ ˆ x y z x x y y z z x x y y z z x y z i j k B B A C A C A C B A B A B A B C C C C × = + + + + K K K ( ) ˆ x x y x y z x z x x x y y x z z x C A B C A B C A B C A B C A B C i = + + ( ) ˆ x y y y z y z x x y y y y z z y C A B C A B C A B C A B C A B C j + + + ( ) ˆ x y z y z z z x x z y y z z z z C A B C A B C A B C A B C A B C k + + + ) A B x A B x y A B x z A B ( ) ( ) ( ) ˆ ˆ ˆ y x y z x z y y x z z x x y x z y z x x y z z y x z x y z y x x z y y z A B C A B C A B C A B C i A B C A B C A B C A B C j A B C A B C A B C A B C k + = + + + + 3

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ˆ ˆ ˆ x y y z z y z x x z x y y x i j
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