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, $$\vec{AO} = 2\hat{i} + 3\hat{j} - 4\hat{k}$$

We need to find the vector sum of the displacements from vertex $$A$$ to all other vertices of a regular octagon $$ABCDEFGH$$ in terms of the position vector of its center $$O$$,

1. Express Vector Formulations relative to the Center ($$O$$)

Let the center of the regular octagon be chosen as the reference origin $$O$$. The position vector of any vertex $$X$$ can be denoted as $$\vec{OX}$$. By using triangle law of vector addition, any vector starting from $$A$$ can be written as:

$$\vec{AX} = \vec{AO} + \vec{OX}$$

We need to find the sum ($$\vec{S}$$):

$$\vec{S} = \vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} + \vec{AG} + \vec{AH}$$

2. Substitute with Geometric Vectors

Expanding each term relative to the center $$O$$:

$$\vec{AB} = \vec{AO} + \vec{OB}$$

$$\vec{AC} = \vec{AO} + \vec{OC}$$

$$\vec{AD} = \vec{AO} + \vec{OD}$$

$$\vec{AE} = \vec{AO} + \vec{OE}$$

$$\vec{AF} = \vec{AO} + \vec{OF}$$

$$\vec{AG} = \vec{AO} + \vec{OG}$$

$$\vec{AH} = \vec{AO} + \vec{OH}$$

Adding all 7 equations together:

$$\vec{S} = 7\vec{AO} + (\vec{OB} + \vec{OC} + \vec{OD} + \vec{OE} + \vec{OF} + \vec{OG} + \vec{OH})$$

3. Utilize Symmetry of a Regular Octagon

For a symmetrical regular polygon, the vector sum of the position vectors of all vertices measured from the geometric center $$O$$ is always zero:

$$\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} + \vec{OE} + \vec{OF} + \vec{OG} + \vec{OH} = \vec{0}$$

From this geometric identity, we can isolate the sum of the remaining 7 center-to-vertex vectors:

$$\vec{OB} + \vec{OC} + \vec{OD} + \vec{OE} + \vec{OF} + \vec{OG} + \vec{OH} = -\vec{OA}$$

Since the vector pointing from the center to vertex $$A$$ ($$\vec{OA}$$) is equal and opposite to the vector pointing from $$A$$ to the center ($$\vec{AO}$$), we have $$-\vec{OA} = \vec{AO}$$. Substituting this value back into our summation expression:

$$\vec{S} = 7\vec{AO} + \vec{AO} = 8\vec{AO}$$

4. Compute Component Multiplication

We are given the coordinate vector values for $$\vec{AO}$$:

$$\vec{AO} = 2\hat{i} + 3\hat{j} - 4\hat{k}$$

Multiplying the vector by the scalar factor of 8:

$$\vec{S} = 8 \times (2\hat{i} + 3\hat{j} - 4\hat{k})$$

$$\vec{S} = 16\hat{i} + 24\hat{j} - 32\hat{k}$$

Correct Option: A ($$16\hat{i} + 24\hat{j} - 32\hat{k}$$)