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vishal_kothari

  • 2 years ago

A,B,C,D are four distinct points in three space. Suppose each of the angles ABC, BCD, CDA, and DAB are right angles. Show that all four points lie in the same plane.

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  1. badreferences
    • 2 years ago
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    Is heuristic reasoning okay, or should I make a rigorous proof?

  2. vishal_kothari
    • 2 years ago
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    easy proof...

  3. badreferences
    • 2 years ago
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    Not for me. :P

  4. vishal_kothari
    • 2 years ago
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    yes for u only..

  5. AravindG
    • 2 years ago
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    this is easy

  6. AravindG
    • 2 years ago
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    tell me which text u refer

  7. vishal_kothari
    • 2 years ago
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    yeah...

  8. AravindG
    • 2 years ago
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    do u have balagopal book?

  9. vishal_kothari
    • 2 years ago
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    yeah..

  10. AravindG
    • 2 years ago
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    this is there in it!!

  11. AravindG
    • 2 years ago
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    refer page 116

  12. vishal_kothari
    • 2 years ago
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    ya but the proof is very easy....

  13. badreferences
    • 2 years ago
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    Proof by probabilistic assumption. :P There are no sets of regular non-planes that adhere to the rules you specify.

  14. vishal_kothari
    • 2 years ago
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    kkk

  15. badreferences
    • 2 years ago
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    Lol, is that really an answer? Laaame.

  16. vishal_kothari
    • 2 years ago
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    no...it needs 3 d geometry....

  17. vishal_kothari
    • 2 years ago
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    what are u typing?

  18. badreferences
    • 2 years ago
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    A rigorous mathematical proof by extending vectors. XD

  19. badreferences
    • 2 years ago
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    Not including non-Euclidian, of course, because I don't know that stuff.

  20. Ishaan94
    • 2 years ago
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    I am not sure about this, It's been time since I have done 3d. If you can take A,B,C,D to be following vectors \(\vec{a},\vec{b},\vec{c} \) and \(\vec{d}\), and then apply the dot product rule.

  21. AravindG
    • 2 years ago
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    use balagopal book

  22. Ishaan94
    • 2 years ago
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    \[AB = \vec{b} - \vec{a}\]\[BC= \vec{c} - \vec{b}\] \[\angle ABC=90 \implies (\vec{b} - \vec{a})(\vec{c} - \vec{b})=0 \] Something like this, and then applying the same procedure for every angle. In the end you might get something.

  23. badreferences
    • 2 years ago
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    Gah, I wish I had a tablet.

  24. vishal_kothari
    • 2 years ago
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    take it then...

  25. vishal_kothari
    • 2 years ago
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    bye..

  26. badreferences
    • 2 years ago
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    lol Ishaan has the right idea. We take what we know... the vector sum being zero, and each individual vector extending in another component from the previous, alternating and opposite. We can determine then, through mathematics that I tried HTML typing, that the only possible unit vector components are ones that cancel out.

  27. badreferences
    • 2 years ago
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    We can probably determine the last part through annoying set elimination by contradiction or through matrix algebra.

  28. vishal_kothari
    • 2 years ago
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    thanks....

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