Coordinates of Hexagon Vertices in Base (AC, AD)

[Petrus]

Consider a regular hexagon ABCDEF (in order counterclockwise). Determine the coordinates of AB, AE AND AF (->) in the base (AC, AD) (->)

AB(->)=(_____,_____)
AE(->)=(_____,_____)
AF(->)=(_____,_____)

what I mean with exemple AF(->) positive way from A to F. I have draw a it but I got problem to rewrite
I got AE=EF+FA(->) I am correct?

 

[Petrus]

Re: hexagon, coordinates, base

I could uppload picture from internet but it should be insted of F it should B. In ourder counterclockwise. Well I can rewrite AE=EF+FA (->)

 

[HallsofIvy]

Re: hexagon, coordinates, base

No, you have the order wrong: AE= AF+ FE.

 

[Petrus]

Re: hexagon, coordinates, base

No, you have the order wrong: AE= AF+ FE.

Yeah I forgot to Edit. How do I do for AF and AB? Then I got Also My base.

 

[Petrus]

Re: hexagon, coordinates, base

View attachment 725
So I draw it.
We know from origo to A it is 1 and origo to D it is 1.

 

[Petrus]

I think I have thought wrong...
We got the base AC and AD and I put $$AC=(1,0)$$ and $$AD(0,1)$$
I start with AB
We know that $$AC=AB+BC$$ ( ->) that means $$AB=AC-BC <=> AB=AC-BC$$ and we know that $$AC=(1,0)$$ That means $$AB=(1,0)-BC$$ But is BC same as from A to origo?

 

[Petrus]

So I did wrong... After a lot reading I think I got correct progress now...
We know it says regular hexagon that means from origo to any point got the length 1.
A circle is 360 degree and we got 8 lines. $$\frac{360}{8}=45$$ So we got now (look picture). we know x value is cos and y value is sin so we know
$$A=(1,0)$$
$$B=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$$
$$C=(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$$
$$D=(-1,0)$$
$$E=(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$$
$$F=(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$$
That means
$$AC=(-\frac{1}{\sqrt{2}}-1,\frac{1}{\sqrt{2}})$$
$$AD=(-2,0)$$
----
$$AB=(\frac{1}{\sqrt{2}}-1,\frac{1}{\sqrt{2}})$$
$$AE=(-\frac{1}{\sqrt{2}}-1,-\frac{1}{\sqrt{2}})$$
$$AF=(\frac{1}{\sqrt{2}}-1,-\frac{1}{\sqrt{2}})$$
So now I got all point but got problem to determine our cordinate with our base. View attachment 727