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anonymous

  • one year ago

Do the three lines 2x1 + 4x2 + 4x3 = 4, x2 - 2x3 = -2, and 2x1 + 3x2 = 0 have at least one common point of intersection? It's linear algebra, and has to do with determining the consistency of systems. Like a matrices problem, the 1 2 and 3 represent rows, not exponents.

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  1. rvc
    • one year ago
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    do you know matrices?

  2. tkhunny
    • one year ago
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    Have you considered a determinant?

  3. anonymous
    • one year ago
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    Hi rvc, I know a little about matrices since we just started going over them, but this one has thrown me for a loop.

  4. rvc
    • one year ago
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    equations are consistent if the determinant equals to zero

  5. rvc
    • one year ago
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    hello :)

  6. anonymous
    • one year ago
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    Okay, so how do I make that happen? I was sick and missed the lecture, so I'm a bit lost. I sorry for bothering you. -_-;

  7. rvc
    • one year ago
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    do you remember how to find determinant?

  8. anonymous
    • one year ago
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    No, not really. I'm sorry.

  9. anonymous
    • one year ago
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    I missed it. How do I find the determinant?

  10. rvc
    • one year ago
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    |dw:1441072730208:dw|

  11. anonymous
    • one year ago
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    Hm... So how does that translate into my word problem?

  12. anonymous
    • one year ago
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    if we break it down into a matrices 2 4 4 4 0 1 2 -2 2 3 0 2

  13. anonymous
    • one year ago
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    But I'm not quite sure I understand how the other numbers within the matrices are broken up to be multiplied(?) by either a11 a12 or a13

  14. rvc
    • one year ago
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    atm i need to go. i have my exam in the next hour. all the best and sorry to leave you in between this problem. i m sure @mathmate will help you. :)

  15. anonymous
    • one year ago
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    @tkhunny

  16. dan815
    • one year ago
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    hey have you figured out how to take a determinant yet

  17. anonymous
    • one year ago
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    Nope, I have not. Do you have any time to help me figure out how to do it?

  18. dan815
    • one year ago
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    yeah

  19. anonymous
    • one year ago
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    Wow! That'd be great. I'm trying to work through the problems in the book for the lecture I missed but I'm kind of lost.

  20. dan815
    • one year ago
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    okay basically to start off, you have an equation which is a linear function of 3 variables

  21. dan815
    • one year ago
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    for example 1x+2y+3z=5 lets say this equation instead of x1,x2,x3

  22. anonymous
    • one year ago
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    Alright

  23. dan815
    • one year ago
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    have you seen equations like this before its the qeuation of a plane in 3d space

  24. anonymous
    • one year ago
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    Yes, a long time ago. It's been a while since I've had to work through anything like this though.

  25. dan815
    • one year ago
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    okay i see well dont worry

  26. dan815
    • one year ago
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    now how about a gradient have you heard about gradients or normal vectors to planes

  27. anonymous
    • one year ago
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    Nope. I have not dealt with that. We're just starting out the school year with matrices.

  28. dan815
    • one year ago
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    okay well then for now i cant fully explain to you why this matrix method is working, but lets just go through the determinant

  29. dan815
    • one year ago
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    a non zero determinant means that all 3 of your systems are independant of each other meaning they are not parallel or one of the system is not a combination of the 2 remaining systems

  30. anonymous
    • one year ago
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    That's okay! I appreciate your willingness to explain!

  31. anonymous
    • one year ago
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    That makes sense.

  32. dan815
    • one year ago
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    okay we dont really care about the constant term for now, again this has something do with the normal vectors how they wont depend on the constant

  33. dan815
    • one year ago
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    so your matrix is the constants of all the coefficceints in order

  34. anonymous
    • one year ago
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    Alright. So we break it up into a matrix... 2 4 4 4 0 1 2 -2 2 3 0 2

  35. dan815
    • one year ago
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    |dw:1441074412998:dw|

  36. dan815
    • one year ago
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    we dont care about the constants in this case

  37. anonymous
    • one year ago
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    Alright, so just the first three columns then.

  38. dan815
    • one year ago
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    yeah the numbers infront of variables only

  39. anonymous
    • one year ago
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    2 4 4 0 1 2 2 3 0

  40. dan815
    • one year ago
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    now to take a determinant of a matrix

  41. dan815
    • one year ago
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    first u learn to take the determinant of 2 by 2 matrix

  42. anonymous
    • one year ago
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    How do you take the determinant.

  43. dan815
    • one year ago
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    |dw:1441074586338:dw|

  44. dan815
    • one year ago
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    this is the determinnat of 2 by 2 matrix

  45. dan815
    • one year ago
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    difference* of the product of the diagonals in that order

  46. dan815
    • one year ago
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    we will use this to take the determinant of 3by3 matrix

  47. anonymous
    • one year ago
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    Okay. How would you combine a 3 by 3 though? A B C D E F G H I... _> AE-BD, BF-CE, DH-EG, EI-FH?

  48. dan815
    • one year ago
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    |dw:1441074679006:dw|

  49. dan815
    • one year ago
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    do get the mini determinants from the big determinant what you do is cover this way

  50. dan815
    • one year ago
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    |dw:1441074913573:dw|

  51. dan815
    • one year ago
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    |dw:1441074951428:dw|

  52. dan815
    • one year ago
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    https://www.youtube.com/watch?v=ROFcVgehEYA

  53. dan815
    • one year ago
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    just watch that its a lot better

  54. dan815
    • one year ago
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    since the determinant is non zero you can conclude there will be only 1 intersection as they are all independant

  55. dan815
    • one year ago
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    because if u look at 3 planes

  56. anonymous
    • one year ago
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    Wow... Thank you so much! That's amazing! It makes a lot of sense, way more than I thought it would. This is a great addition to my notes!

  57. dan815
    • one year ago
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    |dw:1441075204298:dw|

  58. dan815
    • one year ago
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    now if u throw in another plane, that means u are really checking the intersection of a line and a plane

  59. dan815
    • one year ago
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    so if that plane is independant that means it will be a line passing through a plane so just a point

  60. dan815
    • one year ago
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    if the plane was not independant that means the line also belongs in that plane, which means the plane stated is actually redundant which is why ud get a 0 determinant, that wont really make sense yet but ya something to keep in mind for later

  61. anonymous
    • one year ago
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    Okay!

  62. dan815
    • one year ago
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    just know the formal definition of independance

  63. dan815
    • one year ago
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    wait

  64. dan815
    • one year ago
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    http://prntscr.com/8b85p2

  65. anonymous
    • one year ago
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    Okay, what else would you like to explain. I'm not leaving until you're finished ^_^. I just "fanned" you and wrote what I hope will be a glowing testimonial of your efforts.

  66. dan815
    • one year ago
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    basically this a bunch of vectors are said to be linearly independant, if there is no way to do add them or some factor of them up together to end up with 0

  67. anonymous
    • one year ago
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    I will make sure to add the formal definition of independence in my notes as well, so that I remember it for the next lecture.

  68. dan815
    • one year ago
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    for example v1,v2 linearly independant if av1+bv2=0 only happens when both a and b are 0 which is the trivial case, or basically taking nothing to get nothing

  69. anonymous
    • one year ago
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    That makes sense.

  70. dan815
    • one year ago
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    yep basically the definition in there is the same thing not stated for all sizes of vectors

  71. dan815
    • one year ago
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    you can have v1,v2,v3...vn and they all independant if a1*v1+a2*v2+..._an*vn=0 the only solution is that a1,a2,a3...an are all 0

  72. dan815
    • one year ago
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    if that is the only solution, meaning taking nothing to get nothing, everything else doesnt product 0 that means they are lienarly independant

  73. dan815
    • one year ago
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    it is is easy to see somthing like this when u talk about orthogonal vectors which are vectors than are as indepednant as possible of each other

  74. dan815
    • one year ago
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    for example if u take a vector in the x direction and another vector in the y direction, theres no way to add them to get 0,

  75. dan815
    • one year ago
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    once u have the existance of a vector in x, how can you even add some vector in the y direction to make it go back to 0

  76. anonymous
    • one year ago
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    Neat! Thank you!

  77. dan815
    • one year ago
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    you're welcome

  78. dan815
    • one year ago
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    good luck with this stuff

  79. anonymous
    • one year ago
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    Thank you! If nothing else, this experience has taught me to never get sick ever again. That will be the day lecture will be over something I don't understand.

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