Q.OABCDE is a regular hexagon. The points A and B have position vectors $\bar a$ and $\bar b$ respectively, referred to the origin O. Find, in terms of $\bar a$ and $\bar b$, the position vectors of C, D and E.

Set up coordinates with the hexagon's centre at the origin, vertices at $60^\circ$ steps, then shift so that

vertex $O$ becomes the new origin. Writing the position vectors of $A$ and $B$ (relative to $O$) as $\bar a$

and $\bar b$, and solving for the coordinates of $C,D,E$ in terms of $\bar a,\bar b$ using this rotational

structure (each vertex is obtained from the previous by rotating the edge direction by the hexagon's exterior

angle of $60^\circ$) gives:

These can be checked for self-consistency: since $O$ and $C$ are opposite vertices of the hexagon 3 steps

apart, and likewise $A,D$ and $B,E$ are opposite pairs, the three "long diagonals" $OC, AD, BE$ must all share …

Model the regular hexagon with coordinates (circumradius = side length), express every vertex as a linear combination of the given a, b, and cross-check using …

Assuming OB is a side of the hexagon (it is actually a shorter diagonal, since O and B are two vertices apart) leads to the wr …