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Math Help - rectangular coordinate system

  1. #1
    Member Veronica1999's Avatar
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    rectangular coordinate system

    I don't understand this problem but i am still working on it....
    Attached Thumbnails Attached Thumbnails rectangular coordinate system-1981amc.png  
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: rectangular coordinate system

    Hi Veronica!

    It seems to me that the right answer is missing...

    But to find out, suppose the line is of the form \mathbf{a} + \lambda \mathbf{b} with \lambda > 0.
    What are the options for a_x?
    And what are the corresponding options for b_x?
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  3. #3
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    Re: rectangular coordinate system

    Quote Originally Posted by ILikeSerena View Post
    It seems to me that the right answer is missing...
    But to find out, suppose the line is of the form \mathbf{a} + \lambda \mathbf{b} with \lambda > 0.
    What are the options for a_x?
    And what are the corresponding options for b_x?
    I agree that the right answer is not there,
    But what is the right question?
    As is, its wording is extremely odd.
    It seems to me the question is about line segments made of exactly four points taken from a 3D lattice where the points are the 64 latice points, from (1,1,1)\text{ to }(4,4,4).
    We have three sets of four planes with ten lines in each plane.
    BUT I don't know how we should count the diagonals.
    Any thoughts?
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: rectangular coordinate system

    Quote Originally Posted by Plato View Post
    I agree that the right answer is not there,
    But what is the right question?
    As is, its wording is extremely odd.
    It seems to me the question is about line segments made of exactly four points taken from a 3D lattice where the points are the 64 latice points, from (1,1,1)\text{ to }(4,4,4).
    We have three sets of four planes with ten lines in each plane.
    BUT I don't know how we should count the diagonals.
    Any thoughts?
    Umm... I don't see any ambiguity in the question.
    It asks for lines and not for line segments.
    I interpret it to ask for all possible lines with 4 distinct points on it with integer coordinates between 1 and up to 4 (your 64 lattice points).
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    Re: rectangular coordinate system

    Quote Originally Posted by ILikeSerena View Post
    Umm... I don't see any ambiguity in the question.
    It asks for lines and not for line segments.
    I interpret it to ask for all possible lines with 4 distinct points on it with integer coordinates between 1 and up to 4 (your 64 lattice points).
    But that is exactly the same set of lines in either case.
    What about diagonals? Are there more than four?
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    Member Veronica1999's Avatar
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    Re: rectangular coordinate system

    All I know is the answer is (D) 76.
    And I am still trying to understand the problem...
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  7. #7
    Super Member ILikeSerena's Avatar
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    Re: rectangular coordinate system

    Quote Originally Posted by Plato View Post
    But that is exactly the same set of lines in either case.
    What about diagonals? Are there more than four?
    I don't understand what puzzles you...
    In my interpretation there are 4 3D-diagonals, and a number more of 2D-diagonals.


    Quote Originally Posted by Veronica1999 View Post
    All I know is the answer is (D) 76.
    And I am still trying to understand the problem...
    In my count I'm not getting 76 lines.
    Actually, I'm getting 61 lines.
    (And you didn't answer my question.)

    Edit: Hold on, I made a mistake counting...
    Yep, I'm getting 76 lines now.
    Last edited by ILikeSerena; May 25th 2012 at 02:28 PM.
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  8. #8
    Super Member ILikeSerena's Avatar
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    Re: rectangular coordinate system

    Plato had a good point, counting the diagonals.
    How many 3D diagonals are there?
    How many 2D diagonals?
    And how many other lines that have 4 distinct points on them with integer coordinates between 1 and up to 4?
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  9. #9
    Member Veronica1999's Avatar
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    Re: rectangular coordinate system

    Thanks to everyone's help, I now understand the problem.
    But i seem to get 73 lines.
    Attached Thumbnails Attached Thumbnails rectangular coordinate system-ijk.jpg  
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  10. #10
    Super Member ILikeSerena's Avatar
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    Re: rectangular coordinate system

    Hmm... how are you counting?
    For starters, how many lines do you get parallel to the x-axis?
    Did you count the lines that are going through the middle of the cube (and not through any corner or edge)?
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    Re: rectangular coordinate system

    Quote Originally Posted by ILikeSerena View Post
    Hmm... how are you counting?
    For starters, how many lines do you get parallel to the x-axis?
    Did you count the lines that are going through the middle of the cube (and not through any corner or edge)?
    Let me set forth how I see it. Consider a 2D\text{ 4-lattice}.

    We can consider that as a plane parallel to the yz-plane. All coordinates look like (1,j,k),~1\le j,k\le 4.
    There are ten of our lines in that plane: four parallel to the y-axis; four parallel to the z-axis; and two diagonals.
    There are three copies of that plane, (x,j,k),~1\le j,k\le 4~\&~x=2,~3,~4, all parallel to the yz-plane.
    Now there are forty of our lines.

    There are two more sets of those planes: 4 parallel to the xz-plane and 4 parallel to the xy-plane.

    Now we are up to 120. We have 4 diagonals.
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  12. #12
    Super Member ILikeSerena's Avatar
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    Re: rectangular coordinate system

    Quote Originally Posted by Plato View Post
    Let me set forth how I see it. Consider a 2D\text{ 4-lattice}.

    We can consider that as a plane parallel to the yz-plane. All coordinates look like (1,j,k),~1\le j,k\le 4.
    There are ten of our lines in that plane: four parallel to the y-axis; four parallel to the z-axis; and two diagonals.
    There are three copies of that plane, (x,j,k),~1\le j,k\le 4~\&~x=2,~3,~4, all parallel to the yz-plane.
    Now there are forty of our lines.

    There are two more sets of those planes: 4 parallel to the xz-plane and 4 parallel to the xy-plane.

    Now we are up to 120. We have 4 diagonals.
    You are counting lines more than once.
    If we only look at lines parallel to the x-axis, there are 16.
    Times 3 is 48 (and not 96).
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  13. #13
    Member Veronica1999's Avatar
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    Re: rectangular coordinate system

    I finally got 72.
    But is there a more logical approach?
    And less time consuming?
    Where am I missing 4 lines?
    Attached Thumbnails Attached Thumbnails rectangular coordinate system-ijk1.jpg  
    Last edited by Veronica1999; May 25th 2012 at 05:18 PM.
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  14. #14
    Member Veronica1999's Avatar
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    Re: rectangular coordinate system

    I got 76. Is my answer correct now?
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  15. #15
    Super Member ILikeSerena's Avatar
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    Re: rectangular coordinate system

    Hmm, it seems you counted 5 lines everywhere, where you should have counted 4.
    It seems you included coordinate zero, but your problem statement says "positive" integers.

    So for instance the diagonals you drew are actually 3x(4+4)=24.
    Parallel to the axis you have 4x4 lines that intersect a face of your cube, which gives 3x4x4=64 lines.
    Then your final 4 3D-lines gives 24+64+4=76.



    Here's another way.
    Suppose you represent each line by \mathbf{a} + \lambda \mathbf{b} with \lambda \ge 0.
    Then \lambda must take the values 0,1,2,3 to make sure all coordinates are integers.

    Then a_x is either 1,2,3, or 4.
    And b_x can be zero in all cases.
    If a_x=1 then b_x can also be 1.
    And if a_x=4 then b_x can also be -1.

    So there are 6 possibilities.
    The same for y and z gives you 6^3 possibilities.

    But b cannot be the zero vector, because then you would not have distinct points.
    So 4^3 possibilities are wrong.

    This gives you 6^3 - 4^3 possibilities.
    Since you count every line double, you need to divide this result by 2...
    Last edited by ILikeSerena; May 25th 2012 at 11:05 PM.
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