A hexagon has points ABCDEF. Express the position vectors of C,D,E,F in terms of a and b

Question

The origin 0 is the center of a regular hexagon ABCDEF. The vertices A and B have potition vectors a and b. Express the position vectors of C, D, E, and F in terms of a and b.

 

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Answer 1

Let the side of the hexagon be length r, which is also the radius of the circle on which it is inscribed. The 6 vertices can be represented by the following coordinates:

A r(1,0); B r(1/2,sqrt(3)/2); C r(-1/2,sqrt(3)/2); D r(-1,0); E r(-1/2,-sqrt(3)/2); F r(1/2, -sqrt(3)/2).

The unit vectors for x and y are i and j, so the vectors are A=ri and B=(r/2)i+(rsqrt(3)/2)j. So it's clear that D=-A=-a and E=-B=-b. A-B=a-b=(r/2)i-(rsqrt(3)/2)j=F and C=-F=b-a.

 

 

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