Question

In an octagon ABCDEFGH of equal side, what is the sum of $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} + \vec{AG} + \vec{AH}$, if $\vec{AO} = 2i + 3j - 4k$?

• A. 16i + 24j - 32k
• B. -16i - 24j - 32k
• C. 16i + 24j + 32k
• D. 16i + 24j - 32k

Hint:

In a regular polygon with $n$ vertices and center $O$, the sum of vectors from one vertex to all other vertices is always equal to $n \cdot \vec{AO}$.

Answer 1

The Correct Option is A

Step 1: Let $O$ be the center of the regular octagon. The sum of vectors from the center to all vertices is zero: $\sum \vec{OX} = 0$.
Step 2: Express each vector in terms of the center $O$: $\vec{AB} = \vec{OB} - \vec{OA}$, $\vec{AC} = \vec{OC} - \vec{OA}$, etc.
Step 3: Total Sum $= (\vec{OB} + \vec{OC} + \vec{OD} + \vec{OE} + \vec{OF} + \vec{OG} + \vec{OH}) - 7\vec{OA}$.
Step 4: Since $\sum \vec{OX} = 0$, the sum of the other 7 vectors is $-\vec{OA}$.
Step 5: Total Sum $= -\vec{OA} - 7\vec{OA} = -8\vec{OA} = 8\vec{AO}$.
Step 6: $S = 8(2i + 3j - 4k) = 16i + 24j - 32k$.