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Math Help - Help with pythagoras and vector cube problem

  1. #1
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    Help with pythagoras and vector cube problem

    Hi all, I'm new so I hope this is in the right place.

    I have a question about a cube that is sunk unevenly and partway into the ground. I'm then given two points on the cube which are called A and C. The coordinates for these points are:A = (13.0492, 30.9498, 9.01115) and C = (1.97687, 4.68868, 17.3632). I'm then told that the x-y plane lies level with the ground and that points B and D are of equal height and that all measurements are in meters.

    From there I need to find the coordinates for B and D, the side length of the cube, the volume of the cube above and below ground and the coordinates for A', B', C' and D' which are points below their respective letters that intersect with the ground.

    Now I've managed to find several things. Firstly I know the side length is 21m and that points B and D are: B = (21.1958, 12.0502, 13.1872)and D = (-6.16975, 23.5882, 13.1872). But I have no idea on how to solve for points A', B', C' and D'. I was trying to figure it out by forming a right angle triangle from point A to the ground so I'd know the top point is A, the point directly below would be A but with a 0 for the z value and the side length here would be 9.01115m but that's as far as I can get. Can anyone else help me get part this point because I'm not even sure if I'm approaching this right.
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  2. #2
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    I feel like I'm missing something here. Have you stated all the information in the problem statement? If so, can you show me how you got the side length as well as B and D? I'm thinking there have to be additional conditions on all these points!
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  3. #3
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    To find the side length I used pythagoras a^2+b^2=c^2, and since it was a cube a and b are equal, therefore √(c^2/2)=a. I took the distance from A to C as c^2. So it came out as |A-C|=29.6985^2/2=√441=21m.

    To find B and D I first found the z value for them by using the mid-point between A and C. The height difference between the points A and C was 17.3632-9.01115=8.35205. Then this value divided by the distance between A and C gives how far in m the distance falls from C to A per meter, so 8.35205/29.6985=0.28123m. Then multiply that by half way between C and A gives 0.28123X(29.6985/2)=4.1760m. Than either the z value of C-this value or the z value of A+this value gives 13.1872m.

    Then using this value I subbed them into the equation |R-A|^2=s^2 and |R-C|^2=s^2 where R=(x, y, z) and s is the side length. Then subbing our z value in and solving for x and y gave me my coordinates for B and D.

    I'm pretty sure I've included all the info, but just in case I've included the entire question which is a bit to read through.

    It is supposed to be a cube that is sticking out of the ground. Let’s call it the Cube. The
    right photo shows what you see when you zoom in on the Cube from above using Google
    Earth (Google Earth). The left photo was taken from the ground. In
    this photo the corner of the Cube visible at the top is the corner labeled A in the satellite
    photo.
    Google Earth has a feature that allows you to get a 3D view of prominent buildings in
    various cities (check out the Eifel tower in Paris, for example). Imagine that you are
    the person in charge of creating a 3d image of the Cube for Google Earth. For this
    you need to find 3d coordinates of the corners of the top square A,B,C,D, as well as
    the coordinates of the points A0,B0,C0,D0 at which four edges of the Cube intersect the
    ground. Here A is supposed to be connected to A0 by such an edge (see the left photo)
    and similarly for B and B0, C and C0, and D and D0.
    You are not the first person to attempt finding these coordinates. In fact, the last person
    who tried just got fired yesterday because he was not able to do the job. You’ve just
    taken over and you are supposed to base your work on your predecessors notes.
    His notes are a mess, but it is clear that he started by letting the ground coincide with
    the x, y-plane and that the y-axis is supposed to point West and the x-axis North. It is
    not clear where exactly the origin of his coordinate system is, but wherever it is, in this
    coordinate system A = (13.0492, 30.9498, 9.01115) and C = (1.97687, 4.68868, 17.3632).
    You may assume that B and D are at equal height above the ground. Finally, it is also
    clear that all these numbers are measurements in meters. Off you go!
    Wait, apart from finding those eight coordinates I would also like you to figure out how
    long one of the sides of the Cube is; what the lengths of the segments AA0,BB0,CC0,DD0
    are; what the area of the quadrilateral A0,B0,C0,D0 is; and what the volumes of the parts
    of the Cube above and below the ground are.
    10-May-2010 4
    School of Mathematical Sciences Monash University
    At the end of this question please summarize your results as follows:
    A = (13.0492, 30.9498, 9.01115)
    B = (?, ?, ?)
    C = (1.97687, 4.68868, 17.3632)
    D = (?, ?, ?)
    A0 = (?, ?, ?)
    B0 = (?, ?, ?)
    C0 = (?, ?, ?)
    D0 = (?, ?, ?)
    |AA0| =?m
    |BB0| =?m
    |CC0| =?m
    |DD0| =?m
    sidelength =?m
    area(A0B0C0D0) =?m2
    volume above =?m3
    volume below =?m3
    (50 marks): For any single one of these vectors/values that you get wrong
    2 marks will be deducted. Tough rules: one mistake that implies other
    mistakes will result in every single one of these mistakes being penalized
    in this way—Google Earth only accepts results that work! Please make
    absolutely sure that you double-check your results before you submit your
    report!
    Hint: Since we are dealing with a cube that is full of right angles, to quickly check
    whether your coordinates for the different points make sense, calculate some the distances
    between these points and check that some of the edges defined by them meet at
    right angles (using the dot product).
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  4. #4
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    |A-C|=29.6985^2/2=√441=21m.
    The notation is mixed up, but I agree that a side's length is 21m. I also agree that the z component of both B and D is 13.187175. You can just take the average of the z components of A and C to find this.

    I also agree with your coordinates for B and D. The distances AB, BC, CD, and DA all check out to be 21.

    Now, in trying to find A0, I don't think your approach is going to work well, because you don't know the angles. You might be able to figure those out, but why not use a cross product? You know that AB x AD will be in the direction of the side going from A to A0 (that is, depending on how you've assigned the coordinates of B and D - you might have to take the cross product AD x AB. Just get whichever of those two cross products where the z component is negative - that'll be the correct one.) So let N = AB x AD. If you multiply N by the right scalar, then you know that A + N = A0. Just multiply N by the scalar such that the z-component of A0 = A + N is zero. Then the other two components of A0 will be correct.

    Make sense?
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  5. #5
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    Quote Originally Posted by dazza555 View Post
    To find the side length I used pythagoras a^2+b^2=c^2, and since it was a cube a and b are equal, therefore √(c^2/2)=a. I took the distance from A to C as c^2. So it came out as |A-C|=29.6985^2/2=√441=21m.

    To find B and D I first found the z value for them by using the mid-point between A and C. The height difference between the points A and C was 17.3632-9.01115=8.35205. Then this value divided by the distance between A and C gives how far in m the distance falls from C to A per meter, so 8.35205/29.6985=0.28123m. Then multiply that by half way between C and A gives 0.28123X(29.6985/2)=4.1760m. Than either the z value of C-this value or the z value of A+this value gives 13.1872m.

    Then using this value I subbed them into the equation |R-A|^2=s^2 and |R-C|^2=s^2 where R=(x, y, z) and s is the side length. Then subbing our z value in and solving for x and y gave me my coordinates for B and D.

    I'm pretty sure I've included all the info, but just in case I've included the entire question which is a bit to read through.

    It is supposed to be a cube that is sticking out of the ground. Let’s call it the Cube. The
    right photo shows what you see when you zoom in on the Cube from above using Google
    Earth (Google Earth). The left photo was taken from the ground. In
    this photo the corner of the Cube visible at the top is the corner labeled A in the satellite
    photo.
    Google Earth has a feature that allows you to get a 3D view of prominent buildings in
    various cities (check out the Eifel tower in Paris, for example). Imagine that you are
    the person in charge of creating a 3d image of the Cube for Google Earth. For this
    you need to find 3d coordinates of the corners of the top square A,B,C,D, as well as
    the coordinates of the points A0,B0,C0,D0 at which four edges of the Cube intersect the
    ground. Here A is supposed to be connected to A0 by such an edge (see the left photo)
    and similarly for B and B0, C and C0, and D and D0.
    You are not the first person to attempt finding these coordinates. In fact, the last person
    who tried just got fired yesterday because he was not able to do the job. You’ve just
    taken over and you are supposed to base your work on your predecessors notes.
    His notes are a mess, but it is clear that he started by letting the ground coincide with
    the x, y-plane and that the y-axis is supposed to point West and the x-axis North. It is
    not clear where exactly the origin of his coordinate system is, but wherever it is, in this
    coordinate system A = (13.0492, 30.9498, 9.01115) and C = (1.97687, 4.68868, 17.3632).
    You may assume that B and D are at equal height above the ground. Finally, it is also
    clear that all these numbers are measurements in meters. Off you go!
    Wait, apart from finding those eight coordinates I would also like you to figure out how
    long one of the sides of the Cube is; what the lengths of the segments AA0,BB0,CC0,DD0
    are; what the area of the quadrilateral A0,B0,C0,D0 is; and what the volumes of the parts
    of the Cube above and below the ground are.
    10-May-2010 4
    School of Mathematical Sciences Monash University
    At the end of this question please summarize your results as follows:
    A = (13.0492, 30.9498, 9.01115)
    B = (?, ?, ?)
    C = (1.97687, 4.68868, 17.3632)
    D = (?, ?, ?)
    A0 = (?, ?, ?)
    B0 = (?, ?, ?)
    C0 = (?, ?, ?)
    D0 = (?, ?, ?)
    |AA0| =?m
    |BB0| =?m
    |CC0| =?m
    |DD0| =?m
    sidelength =?m
    area(A0B0C0D0) =?m2
    volume above =?m3
    volume below =?m3
    (50 marks): For any single one of these vectors/values that you get wrong
    2 marks will be deducted. Tough rules: one mistake that implies other
    mistakes will result in every single one of these mistakes being penalized
    in this way—Google Earth only accepts results that work! Please make
    absolutely sure that you double-check your results before you submit your
    report!
    Hint: Since we are dealing with a cube that is full of right angles, to quickly check
    whether your coordinates for the different points make sense, calculate some the distances
    between these points and check that some of the edges defined by them meet at

    right angles (using the dot product).
    The stuff in red, not to mention the question itself, highly suggests that this question is part of an assignment that counts towards your final grade. Please read the forum rules regarding such questions (http://www.mathhelpforum.com/math-he...ng-151418.html). I'm sure Burkard would agree ....

    Thread closed.
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  6. #6
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    Quote Originally Posted by mr fantastic View Post
    The stuff in red, not to mention the question itself, highly suggests that this question is part of an assignment that counts towards your final grade. Please read the forum rules regarding such questions (http://www.mathhelpforum.com/math-he...ng-151418.html). I'm sure Burkard would agree ....

    Thread closed.
    Re-opened after getting pm from OP explaining that the assignment due date had passed before the question was posted.
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