Assertion $$A$$ : If $$A, B, C, D$$ are four points on a semi-circular arc with a centre at $$O$$ such that $$\left|\overrightarrow{AB}\right| = \left|\overrightarrow{BC}\right| = \left|\overrightarrow{CD}\right|$$. Then, $$\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} = 4\overrightarrow{AO} + \overrightarrow{OB} + \overrightarrow{OC}$$
Reason $$R$$ : Polygon law of vector addition yields $$\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} = \overrightarrow{AD} = 2\overrightarrow{AO}$$

In the light of the above statements, choose the most appropriate answer from the options given below.

Analysis of the Assertion:

The assertion states that for four points $$A,\ B, \ C, \ D$$ on a semi-circular arc with center $$O$$ such that $$|\vec{AB}|=|\vec{BC}|=|\vec{CD}|$$:

$$\vec{AB}+\vec{AC}+\vec{AD} = 4\vec{AO} + \vec{OB} + \vec{OC}$$

We can rewrite the vectors on the left side in terms of the origin (center) $$O$$:

1. $$\vec{AB} = \vec{OB} - \vec{OA}$$
2. $$\vec{AC} = \vec{OC} - \vec{OA}$$
3. $$\vec{AD} = \vec{OD} - \vec{OA}$$

Summing these gives,

$$\vec{AB}+\vec{AC}+\vec{AD} = (\vec{OB}+\vec{OC}+\vec{OD}) - 3\vec{OA}$$

We find that, since $$\vec{OA}$$ and $$\vec{OD}$$ are opposite radii originating from $$O$$, $$\vec{OD} = -\vec{OA} = \vec{AO}$$, since $$\vec{OA} = -\vec{AO}$$ (same length but opposite direction), we therefore get,

$$\vec{AB}+\vec{AC}+\vec{AD} = (\vec{OB}+\vec{OC}+\vec{AO}) + 3\vec{AO}$$

or,

$$\vec{AB}+\vec{AC}+\vec{AD} = 4\vec{AO} + \vec{OB} + \vec{OC}$$

Assertion A is true.

Analysis of the Reason:

By the polygon law of vector addition, $$\vec{AB}+\vec{BC}+\vec{CD} = \vec{AD}$$, and we also see that $$\vec{AD} = 2\vec{AO}$$, since $$\vec{AD}$$ is the diameter, and $$\vec{AO}$$ is the radius.

Thus, the reason R is correct.

But the polygon law of vector addition has not been useful in examining the assertion A while we derived equality in assertion A. Thus, although both the assertion A and the reason R are correct, R is not the correct explanation of A.