Math Is Fun Forum

mrpace

Member

Registered: 2012-08-16

Posts: 88

Linear algebra help??

Show that if A and B represent reflections in R3, then AB represents a rotation in R3.

Any help is appreciated.

Last edited by mrpace (2014-09-09 15:46:04)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Linear algebra help??

hi mrpace,

This only happens when the lines of the reflections are not parallel.  (In that case, you get a translation.)

So start by assigning a letter to the point where the lines cross, say C.

Show that C is invariant.

Pick another point , say P.

Show that CP is constant.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Linear algebra help??

What happenes when the lines do not cross and are not parallel? We are in R3.

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“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
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Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Linear algebra help??

hi Stefy,

Thanks for that.  I missed that little detail. 

OK; let's see if this can be dimensioned up. 

A reflection in 3D must be a reflection in a plane.

A rotation in 3D must be about an axis.

If the planes are parallel then translation as in 2D.

If not then they will intersect, in a line.  So that must be the axis of rotation.

Points on the line will be invariant under the two reflections, so that means invariant under a rotation around the line.

Points at a distance from the line will stay at that distance during the reflections .... so it's much the same really.

There are exactly four isometries.  If there is an invariant point and 'handedness' is preserved (clockwise sense, say, as you go from point to point) then the transformation is a rotation.  If the handedness changes it's a reflection.

If there is no invariant point and handedness is preserved, it's a translation.  Otherwise it's a glide reflection (reflection followed by translation in the direction of the line or plane).

As two reflections restore the handedness, the compound transformation must be a rotation or translation.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob