# 64 0 when 90 90 the pipe wrench is horizontal 2 0 0 0

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Unformatted text preview: v2 w3 u3 v1 w1 j u2w3 u1 v2 w2 u2 v1 w1 k w2 i u3v1 j u2v1 k u3w2 i u1w2 u2w1 k 59. u u u u1, u2, u3 ij u1 u 2 u1 u2 v u v v u2v3 u2v3 u2v3 u and u k u3 u3 u2u3 u3u2 i u1u3 u3u1 j u1u2 u2u1 k 0 61. u u u v v u3v2 i u3v2 u1 u3v2 v1 v v. u1v3 u3v1 u3v1 u3v1 j u1v3 u2 u1v3 v2 u1v2 u1v2 u1v2 u2v1 k u2v1 u3 u2v1 v3 0 0 Thus, u 63. u v u v sin If u and v are orthogonal, 2 and sin 1. Therefore, u v u v. Section 10.5 1. x (a) 1 3t, y z Lines and Planes in Space 2 t, z 2 5t (b) When t 0 we have P Q 10, 1, 17 . \ 1, 2, 2 . When t 3 we have PQ 9, 3, 15 \ The components of the vector and the coefficients of t are proportional since the line is parallel to PQ . x y (c) y 0 when t 2. Thus, x Point: 7, 0, 12 x z 0 when t 0 when t 1 . Point: 3 2 . Point: 5 7 and z 71 0, , 33 1 12 , ,0 55 12. 3. Point: (0, 0, 0 Direction vector: v (a) Parametric: x (b) Symmetric: x 1, 2, 3 t, y y 2 z 3 2t, z 3t Direction numbers: 1, 2, 3 5. Point: 2, 0, 3 2, 4, 2 2t, y y 4 z 4t, z 3 2 3 2t 2 2 2 2 Direction vector: v (a) Parametric: x (b) Symmetric: x Direction numbers: 2, 4, Section 10.5 Lines and Planes in Space 22 , ,1 33 17 i 3 11, 17t, y y 3 11 z 5 5 17 11 j 3 9 3 2 9 11t, z 2 3k 245 7. Point: 1, 0, 1 Direction vector: v Direction numbers: 3, (a) Parametric: x (b) Symmetric: x 3 1 1 3i 2j 2, 1 3t, y y 2 z 1 2t, z 1 1 t k 9. Points: 5, 3, 2, Direction vector: v Direction numbers: 17, (a) Parametric: x (b) Symmetric: x 9t 11. Points: 2, 3, 0 , 10, 8, 12 13. Point: 2, 3, 4 Direction vector: v Parametric: x 2, y k 3, z 4 t Direction numbers: 0, 0, 1 3 3 5 z 12 17. Li: v 3, 2, 4 6, 4, 6, 4, 8 6, 4, 6 8 6, 6, 6, 2, 5 on line 2, 5 on line 2, 5 not on line 5t, z 12t Direction vector: 8, 5, 12 Direction numbers: 8, 5, 12 (a) Parametric: x (b) Symmetric: x 8 2 2 8t, y y 15. Point: ( 2, 3, 1 Direction vector: v Parametric: x Symmetric: (a) On line (b) On line (c) Not on line y (d) Not on line 6 4 3 2 2 1 1 x 4 2 2 z 4i k 1 3, z 3 1 t 1 ,y 1 Direction numbers: 4, 0, L 2: v L 3: v L 4: v 4t, y not parallel to L1, L 2, nor L 3 Hence, L1 and L 2 are identical. L1 L 2 and...
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