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# [Phys-L] Re: Parallel lines

On Thursday, Mar 24, 2005, Bob LaMontagne wrote:

All my random numbers were between -1 and +1.
But the error was to think that each of the three
equations defines a line. In reality it defines a
plane, as signaled by Dawid Bowman. Three
random planes in 3D usually define a unique
point but three lines in 3D do not.

Ludwik Kowalski
Let the perfect not be the enemy of the good.

What is the range of the random numbers you are using?

Bob at PC

-----Original Message-----
From: Forum for Physics Educators [mailto:PHYS-L@list1.ucc.nau.edu] On
Behalf Of Ludwik Kowalski
Sent: Thursday, March 24, 2005 11:28 AM
To: PHYS-L@LISTS.NAU.EDU
Subject: Parallel lines

Intuition tells me that two randomly-selected lines, in a plane, will
most likely be non-parallel. In other words they will nearly always
intersect somewhere. This can easily be verified by writing two
equations with two unknowns:

A*x1 + B*x2=K
D*x1 + E*x2=L

and selecting six coefficients randomly. Due to the final accuracy of
computer calculations the term "parallel" must be defined by saying ,
for example, that the absolute value -- of the 2 by 2 determinant of
coefficients -- should be less than 0.001. With that arbitrarily
definition my little program said that 99.8% of the equation sets
described two intercepting lines. In other words, solutions were unique
in nearly all cases. This is not surprising.

But the case of three randomly-selected lines, in 3-dimensional space,
is a surprise. In that case, intuition tells me, randomly-selected
lines will most often have no common point. In other words, unique
solutions should be rare. To verify this I wrote a system of three
linear equations with three unknowns:

A*x1 + B*x2 +C*x3 =K
D*x1 + E*x2 +F*x3 =L
G*x1 + H*x2 +I*x3 =M

and selected their twelve coefficients randomly. The absolute values of
determinants were nearly never very small. In other words, nearly all
set of equations described lines intercepting somewhere. In all trials
random choices were from the uniform distribution between -1 and +1.
The situation is paradoxical. Intuition tells me that unique solutions
should be rare (three lines rarely intercept at a common point) but the
program tells me that unique solutions are not rare. Where am I wrong?
I am nearly certain that programs are error-free. Something must be
wrong with reasoning.

Ludwik Kowalski
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