Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.
Assertion $A$: If $A, B, C, D$ are four points on a semi-circular arc with centre at $O$ such that $|\vec{AB}| = |\vec{BC}| = |\vec{CD}|$, then $\vec{AB} + \vec{AC} + \vec{AD} = 4\vec{AO} + \vec{OB} + \vec{OC}$.
Reason $R$: Polygon law of vector addition yields $\vec{AB} + \vec{BC} + \vec{CD} = \vec{AD} = 2\vec{AO}$.
In the light of the above statements, choose the most appropriate answer from the options given below.
Answer: (D) Both $A$ and $R$ are correct but $R$ is not the correct explanation of $A$.
Take position vectors from $O$: $\vec a, \vec b, \vec c, \vec d$ with $\vec d = -\vec a$ (since $AD$ is a diameter).
**Assertion:** $\vec{AB} + \vec{AC} + \vec{AD} = (\vec b - \vec a) + (\vec c - \vec a) + (\vec d - \vec a) = \vec b + \vec c - 4\vec a$.
Since $\vec{AO} = -\vec a$, $\vec{OB} = \vec b$, $\vec{OC} = \vec c$, this equals $4\vec{AO} + \vec{OB} + \vec{OC}$. So $A$ is true.
**Reason:** By the polygon law, $\vec{AB} + \vec{BC} + \vec{CD} = \vec{AD}$, and $\vec{AD} = 2\vec{AO}$ because $O$ is the midpoint of $AD$. So $R$ is true.
The assertion is proved using the triangle law for each vector separately, not the polygon relation in $R$; hence $R$ is not the correct explanation of $A$.
Solution by Sreeraj P, M.Sc Physics