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Math Help - Rhombus (vectors)

  1. #1
    Mtl
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    Rhombus (vectors)

    If x= 3i -4j-k and y=[3,2,-1], are diagonals of a parrallelagram, then show that this parrarellagram is a rhombus.

    So, I'm assuming I would have to prove that the magnitudes of both y and x are equal. But I'm a bit confused with this unit vector deal. (i,j,k)
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  2. #2
    Member Henderson's Avatar
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    First, the "unit vector deal" you're having problems with is just notation; in your example, x is he same as [3, -4, -1].

    You're not quite right about hoping they have equal magnitutes. A parallelogram with equal-lengthed diagonals is a rectangle, not a rhombus. A rhombus has perpendicular diagonals, so they want you to take the dot product and show that it equals zero.

    Speaking of which, I'm guessing you copied the problem down wrong: either x=3i-4j+k, or y=[3,2,+1].

    Hope this helps.
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  3. #3
    Mtl
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    "Speaking of which, I'm guessing you copied the problem down wrong"

    Yes you are right. The question should read that y=[2,3,-6]
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    Member Henderson's Avatar
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    That'll work out just fine.
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  5. #5
    Mtl
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    Ok, I was confusing the sides with the diagonals. Thanks for your help.
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  6. #6
    Mtl
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    Now, how could I go about trying to find the length of the sides of the rhombus?
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    There is something wrong with the way you have given the problem.

    It is well known that: A parallelogram is a rhombus if and only if its diagonals are perpendicular.
    So see if their dot product is zero.
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  8. #8
    Mtl
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    "There is something wrong with the way you have given the problem.

    It is well known that: A parallelogram is a rhombus if and only if its diagonals are perpendicular.
    So see if their dot product is zero."

    The question asked to prove that it was a rhombus, which Henderson realized could not be done as I gave the wrong numbers to begin with.
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  9. #9
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    Quote Originally Posted by Mtl View Post
    The question asked to prove that it was a rhombus, which Henderson realized could not be done as I gave the wrong numbers to begin with.
    That is the point.
    If the dot product x \cdot y \ne 0 then it is not a rhombus.
    Either the problem is wrong or you have copied it incorrectly.
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  10. #10
    Mtl
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    Ok, I'm still confused here... Wouldn't the diagonals' magnitude have to be equal in a rhombus?
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  11. #11
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    Quote Originally Posted by Mtl View Post
    Ok, I'm still confused here... Wouldn't the diagonals' magnitude have to be equal in a rhombus?
    No, that is true of rectangles. It is not necessary in a rhombus.
    A rhombus with congruent diagonals is a square.
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  12. #12
    Mtl
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    Oh ok... So then to determine the length of the sides of the rhombus I would just divide the magnitudes of the two diagonals by 2 and then use pythagorum's theorum?
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  13. #13
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    Quote Originally Posted by Mtl View Post
    If x= 3i -4j-k and y=[3,2,1], are diagonals of a parallelogram, then show that this parallelogram is a rhombus.
    I have a hard time understanding why you need the length of the sides.
    Note I have changed one sign in the problem. Now x \cdot y = \left\langle {3, - 4, - 1} \right\rangle  \cdot \left\langle {3,2,1} \right\rangle  = 0.
    Now that alone tells us the parallelogram with diagonals x & y is a rhombus.
    There is no need to find the lengths of the sides.
    The problem does not call for it.
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  14. #14
    Mtl
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    Sorry for the confusion, but the original question was indeed copied down wrong by me it should have read:
    If x= 3i - 4j -k and y= [2,3,-6], are the diagonals of a parrallelgram then,
    (a) show that the parralelagram is a rhombus
    (b) find the length of the sides of the rhombus

    *I have already proved part A through a dot product but I'm not sure as to how to figure out part B.
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  15. #15
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    Well, we are not mind readers.
    If s is a side of the rhombus then
    \left\| s \right\|^2  = \left\| {.5x} \right\|^2  + \left\| {.5y} \right\|^2
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