Question 64: Motion in 2D & Vectors

Question

Assertion A : If A, B, C, D are four points on a semi-circular are with 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 :

Accepted Answer

Let the position vectors of $A,B,C,D,O$ be $\vec a,\vec b,\vec c,\vec d,\vec 0$ respectively.
Check Assertion (A)
$ \overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD} =(\vec b-\vec a)+(\vec c-\vec a)+(\vec d-\vec a) =\vec b+\vec c+\vec d-3\vec a $
Also,
$ 4\overrightarrow{AO}+\overrightarrow{OB}+\overrightarrow{OC} =4(\vec 0-\vec a)+(\vec b-\vec 0)+(\vec c-\vec 0) =-4\vec a+\vec b+\vec c $
Equating both sides gives
$ \vec b+\vec c+\vec d-3\vec a=-4\vec a+\vec b+\vec c \;\Rightarrow\; \vec d=-\vec a $
But $\vec d=-\vec a$ means $O$ is the midpoint of $AD$, i.e. $AD$ is a diameter. Since $A$ and $D$ are the endpoints of the given semicircle (as in the figure), this is true.
So
Assertion A is correct
(and it doesnÃ¢ÂÂt actually need $|AB|=|BC|=|CD|$).
Check Reason (R)
By the polygon law along the path $A	o B	o C	o D$,
$ \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}=\overrightarrow{AD} $
Hence
$ \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{AD} =2\overrightarrow{AD}
eq 2\overrightarrow{AO} $
So
Reason R is incorrect
.
Correct option
Option A: A is correct but R is not correct.

Answer

A is correct but R is not correct.

Answer

A is not correct but R is correct.

Answer

Both A and R are correct and R is the correct explanation of A.

Answer

Both A and R are correct but R is not the correct explanation of A.

Motion in 2D & Vectors

Choose the correct answer: