Question

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.

Answer 1

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

Answer 2

easy proof...

Answer 3

Not for me. :P

Answer 4

yes for u only..

Answer 5

this is easy

Answer 6

tell me which text u refer

Answer 7

yeah...

Answer 8

do u have balagopal book?

Answer 9

yeah..

Answer 10

this is there in it!!

Answer 11

refer page 116

Answer 12

ya but the proof is very easy....

Answer 13

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

Answer 14

kkk

Answer 15

Lol, is that really an answer? Laaame.

Answer 16

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

Answer 17

what are u typing?

Answer 18

A rigorous mathematical proof by extending vectors. XD

Answer 19

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

Answer 20

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.

Answer 21

use balagopal book

Answer 22

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

Answer 23

Gah, I wish I had a tablet.

Answer 24

take it then...

Answer 25

bye..

Answer 26

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.

Answer 27

We can probably determine the last part through annoying set elimination by contradiction or through matrix algebra.

Answer 28

thanks....