Need examples of Euclidean isometries f and g such that $\displaystyle f\circ{g}$ ≠ $\displaystyle g\circ{f}$.

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- Dec 13th 2009, 12:58 PMReneePattEuclidean Isometries - Help Please!
Need examples of Euclidean isometries f and g such that $\displaystyle f\circ{g}$ ≠ $\displaystyle g\circ{f}$.

- Dec 13th 2009, 01:45 PMLaurent
- Dec 13th 2009, 02:02 PMReneePatt
- Dec 13th 2009, 07:55 PMKrahl
let A be the matrix [cos90 -sin90|sin90 cos90] a rotation of 90. let X=(x,y) vector.

let f(X)=AX so the action of f on X rotates it by 90.

and let g be a rotation with respect to the point (1,0) not the origin.

so what g does is

g(X)=A(x-1,y)+(1,0) (i.e. take 1 away from x then rotate w.r.t origin then replace x)

so (fog)(X)=f(g(x))=A(A(x-1,y)+(1,0))

and (gof)(X)=g(f(x))=A(AX-(1,0))+(1,0)

picking point (2,1)

fog(X)=(-1,0)

gof(X)=(-1,-2)

so check by putting values for the vector X and seeing these two arent always equal.

((x-1,y) and similar expressions are vectors)