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If vec aa n d vec b are the vectors det...

If ` vec aa n d vec b` are the vectors determined by two adjacent sides of a regular hexagon, what are the vectors determined by the other sides taken in order?

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Let ABCDEF be a regular hexagon
Such that `vec(AB)=vecp`and `vec(BC)=vecq`.
`:.` Here AD is parallel to BC and AD =2BC.
`:. Vec(AD)=2vec(BC)=vec(2q)`. In `Delta`ABC, we have

`vec(AB)+vec(BC)=vec(AC)`
`vecp+vecq=vec(AC)`
Now in `Delta`ACD
`vec(AC)+vec(CD)=vec(AD)`
`rArr vec(CD)=vec(AD)-vec(AC)`
`=2q-(vecp+vecq)`
`=vec q-vecp`
Clearly, `vec(DE)=-vec(AB)=-vec-p`
`vec(EF)=-vec(BC)=-vec-q`
And `vec(FA)=-vec(CD)=vec(p)-vec(q)`
Therefore, vectors determined by other side of a regular hexagon is `vecq-vecp,-vecp,-vecq` and `vecp-vecq`
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