ESE 318-02, Spring 2016 Homework Set #7 Due Tuesday, Mar. 8 1. Zill 7.4.10. 2. Zill 7.4.13. 3. Zill 7.4.24 4. Zill 7.4.25. 5. Zill 7.4.46(b). 6. Write (a) a vector equation, (b) parametric equations and (c) symmetric equations for a line that goes through the two points P1 = (1,-4,0) and P2 = (10,3,-2). 7. Write (a) a vector equation, (b) parametric equations and (c) symmetric equations for a line that goes through the two points P1 = (1,4,-9) and P2 = (10,4,-2). 8. Zill 7.5.22. 9. Zill 7.5.24 and 7.5.26. 10. Zill 7.5.37. 11. Zill 7.5.40. 12. Zill 7.5.46. 13. Consider these two lines that go through a common point. 3,1,15,1,1:2651121:1tLinezyxLine (a) Write an equation of a line that is perpendicular to both Line 1 and Line 2 and goes through that same common point. (b) Write an equation of a plane that contains Line 1 and Line 2. 14. Zill 7.5.57. 15. For each of the following sets, V, determine if the given set is a Vector Space (Subspace) and show your work. If it is a Vector Space, find its dimension and write a basis set. (Each is a subset of a Vector Space, so you only need to check the closure axioms. Also, assume the standard operations of addition and multiplication.) (a) All vectors, 321,, vvvv , in R3, such that 5v1 – 3 v2 + 2 v3 = 0. (b) All 2 X 3 matrices with all non-negative elements. (c) All symmetric 2 X 2 matrices. (d) All skew-symmetric 2 X 2 matrices. (e) All functions  xbxaxf sincos with any constants a and b. 16. V is the set of vectors, 321,, vvvv , in R3, such that v1 + v3 = 0. (Standard operations of addition and multiplication.) (a) Prove V is a Vector Space (Subspace). (Show work.) (b) Find its dimension. (c) Write a basis set. (d) Write the vector 3,7,3v as a linear combination of your basis vectors. 17. Zill 7.6.24(b), 7.6.25 and 7.6.26.ESE 318-02, Spring 2016 1. Zill 7.4.10.      17,86,211668012101021618  kjikjiba 2. Zill 7.4.13. The cross product (and any scalar multiple thereof) is perpendicular to both.      kjiAlsokjikjikjiba523,523724247111472 3. Zill 7.4.24.        1,5,05505252  kjjkikijiiikji 4. Zill 7.4.25.    1,5,5550555  ikjkjijkiiikiji 5. Zill 7.4.46(b). Choose 2 adjacent sides and take the cross product. I’ll use bottom and right here.      28125691616,3,416,3,4164141041411,0,41,4,32,4,11,4,11,4,32,0,2Areakjikjirightbottom 6. Write (a) a vector equation, (b) parametric equations and (c) symmetric equations for a line that goes through the two points P1 = (1,-4,0) and P2 = (10,3,-2). 27491:27491:2,7,90,4,1,,:2,7,92,43,110zyxSymmetrictztytxParametrictzyxVectoraESE 318-02, Spring 2016 7. Write (a) a vector equation, (b) parametric equations and (c) symmetric equations for a line that goes through the two points P1 = (1,4,-9) and P2 = (10,4,-2). 47991:79491:7,0,99,4,1,,:7,0,992,44,110yzxSymmetrictzytxParametrictzyxVectora 8. Zill 7.5.22. 610531261053120zyxtztytx 9. Zill 7.5.24 and 7.5.26. 7.5.24 The point is (4,-11,-7) and the vector is <5,1/3,-2>. 27115431 zyx 7.5.26 A picture helps. For (a), the vector must be parallel to the y axis, so use j. For (b) the vector must be parallel to the z axis, so use k.   tzyxbztyxa821821 10. Zill 7.5.37. Use cross product to find the normal direction. tztytxAlsotztytxkjikjibaba6124,3631643365121115,1,21,1,1 11. Zill 7.5.40.    0202402214  yxoryxoryx 12. Zill 7.5.46. Use the first point, (0,1,0), as the “reference” point for a and b.    12,010221211001,2,11,0,0yxalsoyxjikjibabaESE 318-02, Spring 2016 13. Consider these two lines that go through a common point. 3,1,15,1,1:2651121:1tLinezyxLine (c) Write an equation of a line that is perpendicular to both Line 1 and Line 2 and goes through that same common point. (d) Write an equation of a plane that contains Line 1 and Line 2.           040312305311213:35,121,3135121311,4,15,1,13,12,35,1,1:3,12,31266633116123,1,16,1,2zyxorzyxorzyxPlanetztytxorzyxortortLinekjikjibanormal 14. Zill 7.5.57. Since the lines are parallel, use another vector which goes between two points on those lines.      175790175790157957903517195795121215,1,23,1,12,0,31,2,11,2,1zyxzyxzyxzyxkjikjicacbaESE 318-02, Spring 2016 15. For each of the following sets, V, determine if the given set is a Vector Space (Subspace) and show your work. If it is a Vector Space, find its dimension and write a basis set. (Each is a subset of a Vector Space, so you only need to check the closure axioms. Also, assume the standard operations of addition and multiplication.) (a) All vectors, 321,, vvvv , in R3, such that 5v1 – 3 v2 + 2 v3 = 0. (b) All 2 X 3 matrices with all non-negative elements. (c) All symmetric 2 X 2 matrices. (d) All skew-symmetric 2 X 2 matrices. (e) All functions  xbxaxf sincos with any constants a and b. (a) It is a subset of R3, a Vector Space, so only check closure axioms.           OKwwwvvvwvwvwvzzzwvwvwvwwwvvvzzzzVwvAdditionOKvvvkkvkvkvzzzkvkvkvkvzzzzmultScalar0235235235235,,,,,,,,,0235235235,,,,.321321332211321332211321321321321321321321321    20,5,3,5,0,2:3,2,0,5,0,2,1,0,,0,1:023523252231253321DimensionothersAlsoorBasisvvvvvv (b) Obviously does not satisfy the closure