vishal_kothari
  • vishal_kothari
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.
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
Is heuristic reasoning okay, or should I make a rigorous proof?
vishal_kothari
  • vishal_kothari
easy proof...
anonymous
  • anonymous
Not for me. :P

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vishal_kothari
  • vishal_kothari
yes for u only..
AravindG
  • AravindG
this is easy
AravindG
  • AravindG
tell me which text u refer
vishal_kothari
  • vishal_kothari
yeah...
AravindG
  • AravindG
do u have balagopal book?
vishal_kothari
  • vishal_kothari
yeah..
AravindG
  • AravindG
this is there in it!!
AravindG
  • AravindG
refer page 116
vishal_kothari
  • vishal_kothari
ya but the proof is very easy....
anonymous
  • anonymous
Proof by probabilistic assumption. :P There are no sets of regular non-planes that adhere to the rules you specify.
vishal_kothari
  • vishal_kothari
kkk
anonymous
  • anonymous
Lol, is that really an answer? Laaame.
vishal_kothari
  • vishal_kothari
no...it needs 3 d geometry....
vishal_kothari
  • vishal_kothari
what are u typing?
anonymous
  • anonymous
A rigorous mathematical proof by extending vectors. XD
anonymous
  • anonymous
Not including non-Euclidian, of course, because I don't know that stuff.
anonymous
  • anonymous
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.
AravindG
  • AravindG
use balagopal book
anonymous
  • anonymous
\[AB = \vec{b} - \vec{a}\]\[BC= \vec{c} - \vec{b}\] \[\angle ABC=90 \implies (\vec{b} - \vec{a})(\vec{c} - \vec{b})=0 \] Something like this, and then applying the same procedure for every angle. In the end you might get something.
anonymous
  • anonymous
Gah, I wish I had a tablet.
vishal_kothari
  • vishal_kothari
take it then...
vishal_kothari
  • vishal_kothari
bye..
anonymous
  • anonymous
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.
anonymous
  • anonymous
We can probably determine the last part through annoying set elimination by contradiction or through matrix algebra.
vishal_kothari
  • vishal_kothari
thanks....

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