Two functions f,g : E n E n commute if f(g(P)) = g(f(P)) for all P E n . In othe

Question

Two functions f,g : En En commute if f(g(P)) = g(f(P)) for all P En. In other words, two functions commute if it doesn’t matter which order we have them act on En.

(a) Show that two translations of En always commute.
A fixed point of a function f : En En is a point P such that f(P) = P.

(b) Show that if f and g commute and P is a fixed point of f then g(P) is also a fixed point of f.

For the next problems you may use the following facts with out proof.

-A translation has no fixed points.

-A rotation of E2 has only one fixed point, the center of the rotation.

-The fixed points of a reflection of E2 across a line are the points on the line.

(c) Show that a rotation and a translation never commute.

(d) Show that if a rotation around a point P and a reflection across a line l commute, then p l and the angle of the rotation is 180.

(e) When do a translation and a reflection commute?

Explanation / Answer

(a) Show that two translations of En always commute.

suppose T1 and T2 are two translations with d1 and d2 respectively

let P be arbitary point in En

T1(P) = P+d1 and T2(P) = P + d2

T1(T2(P))= T2(P) + d1

T1(T2(P))= P+ d2 + d1

Now,

T2(T1(P))= T1(P) + d2 .............(1)

T2(T1(P))= P+ d1 + d2   .............(2)

From equation 1 and 2

T2(T1(P))= T2(T1(P))

Since P is arbitary point, T2(T1(P))= T2(T1(P)) for all P in En

So T1 and T2 commutes

(b) Show that if f and g commute and P is a fixed point of f then g(P) is also a fixed point of f.

f(g(P)) = g(f(P)) ..............since f and g commute

f(P) = P ............since P is fixed point of f

applying g on both sides,

g(f(P)) = g(P)

but g and f are commutes. So replating f(g(P)) instead of g(f(P)) we have,

f(g(P)) = g(P)

So this indicates that g(P) is also fixed point of f

(c) Show that a rotation and a translation never commute.

From (b) we know that if f and g commute and P is a fixed point of f then g(P) is also a fixed point of f.

Suppose f is rotation and g is translation

Suppose on contrary, assume that f and g are commutes. Since rotation f has one fixed point as center say C of rotation.

so f(C) = C

but f and g commutes. hence from (b) g(C) is also fixed point of f

but f is rotation so it has unique fixed point hance g(C) is center i.e. C

hence g(C) = C, but it shows that C is fixed point of g but this is contradiction because translation never have fixed point. Hence our assumption of f and g are commutes is wrong. Hence rotation and translation never commutes.

(d) Show that if a rotation around a point P and a reflection across a line l commute, then p l and the angle of the rotation is 180.

Using reflection along line, point is actually mirrored along line. If reflection is applied first, point is mirrored in other side of the line. So it is similar to rotation of point on the by considering center as mid point of original position of point and position of point after reflection by x angle. Also after rotation by it is again by angle y and in reverse way, y and x totally, it has 360 degree rotation and with same degree as 360/3 = 180 degree.

(e) When do a translation and a reflection commute?

If translation of point is on parellel line to reflection line then translation and reflection commutes.

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