Question

\(\text{Assertion A}\) : If \(A,~B,~C,~D\) are four points on a semi-circular arc with centre at \('O'\) such that \(|\overrightarrow {AB}|=|\overrightarrow {BC}|=|\overrightarrow {CD}|\), then \(\overrightarrow {AB}+\overrightarrow {AC}+\overrightarrow {AD}=4~\overrightarrow {AO}+\overrightarrow {OB}+\overrightarrow {OC}\)

\(\text{Reason R}\) : 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 :

• A. $\text{A}$ is not correct but $\text{R}$ is correct.
• B. Both $\text{A}$ and $\text{R}$ are correct and $\text{R}$ is the correct explanation of $\text{A}$
• C. Both $\text{A}$ and $\text{R}$ are correct but $\text{R}$ is not the correct explanation of $\text{A}$.
• D. $\text{A}$ is correct but $\text{R}$ is not correct.

Answer 1

The correct option is C Both $\text{A}$ and $\text{R}$ are correct but $\text{R}$ is not the correct explanation of $\text{A}$.

Given:
$|\overrightarrow{AB}|=|\overrightarrow{BC}|=|\overrightarrow{CD}|$

Here, $O$ is the centre of semi- circle

$\therefore |\overrightarrow{OA}|=|\overrightarrow{OB}|=|\overrightarrow{OC}|=|\overrightarrow{OD}|$

Using vector law of addition, we can write,

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

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

$\overrightarrow{AD}=\overrightarrow{AO}+\overrightarrow{OD}=2\overrightarrow{AO}$

After adding all, we get,

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

$\text{Reason R}$ is the direct result of Polygon law of vector addition

Therefore, Polygon law is applicable in both but the equation given in the reason is not useful in explaining the assertion.

Hence, option $\left(\text{D}\right)$ is correct.