## vishal_kothari Group Title 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. 2 years ago 2 years ago

Is heuristic reasoning okay, or should I make a rigorous proof?

2. vishal_kothari Group Title

easy proof...

Not for me. :P

4. vishal_kothari Group Title

yes for u only..

5. AravindG Group Title

this is easy

6. AravindG Group Title

tell me which text u refer

7. vishal_kothari Group Title

yeah...

8. AravindG Group Title

do u have balagopal book?

9. vishal_kothari Group Title

yeah..

10. AravindG Group Title

this is there in it!!

11. AravindG Group Title

refer page 116

12. vishal_kothari Group Title

ya but the proof is very easy....

Proof by probabilistic assumption. :P There are no sets of regular non-planes that adhere to the rules you specify.

14. vishal_kothari Group Title

kkk

Lol, is that really an answer? Laaame.

16. vishal_kothari Group Title

no...it needs 3 d geometry....

17. vishal_kothari Group Title

what are u typing?

A rigorous mathematical proof by extending vectors. XD

Not including non-Euclidian, of course, because I don't know that stuff.

20. Ishaan94 Group Title

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 Group Title

use balagopal book

22. Ishaan94 Group Title

$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.

Gah, I wish I had a tablet.

24. vishal_kothari Group Title

take it then...

25. vishal_kothari Group Title

bye..

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.