Assertion A : If A, B, C, D are four points on a semi-circular are 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 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 is correct but R is not correct.

A is not correct but R is correct.

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

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

Vector Algebra PYQ — JEE Main 2021 · 27 Jul · Shift 1 · Q58 with Solution

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Correct answer: A

We need to check whether $\vec{AB}+\vec{AC}+\vec{AD}=4\vec{AO}+\vec{OB}+\vec{OC}$ is true.

Using position-vector type decomposition: $\vec{AB}=\vec{AO}+\vec{OB}$ $\vec{AC}=\vec{AO}+\vec{OC}$ $\vec{AD}=\vec{AO}+\vec{OD}$ Hence,

=3\vec{AO}+\vec{OB}+\vec{OC}+\vec{OD}$$ So the assertion will be true if $$3\vec{AO}+\vec{OB}+\vec{OC}+\vec{OD}=4\vec{AO}+\vec{OB}+\vec{OC}$$ which requires $$\vec{OD}=\vec{AO}$$ Now, since $A,B,C,D$ are four points equally spaced on a semicircular arc, the arc is divided into 3 equal parts, so the central angle between consecutive points is $$\frac{180^\circ}{3}=60^\circ$$ Thus the points are at angles $180^\circ,120^\circ,60^\circ,0^\circ$ (for a suitable orientation), so $A$ and $D$ are endpoints of the diameter. Therefore $$\vec{OD}=-\vec{OA}=\vec{AO}$$ Hence the assertion is **correct**. --- 2. **Check the reason** The reason states: $$\vec{AB}+\vec{BC}+\vec{CD}+\vec{AD}=2\vec{AO}$$ Now, $$\vec{AB}+\vec{BC}+\vec{CD}=\vec{AD}$$ by polygon law. Therefore, $$\vec{AB}+\vec{BC}+\vec{CD}+\vec{AD}=2\vec{AD}$$ Since $A$ and $D$ are endpoints of a diameter and $O$ is the midpoint, $$\vec{AD}=2\vec{AO}$$ Therefore, $$2\vec{AD}=4\vec{AO}$$ not $2\vec{AO}$. So the stated reason is **incorrect**. --- 3. **Conclusion** - Assertion A is correct. - Reason R is not correct. Therefore the correct option is: $$\boxed{A}$$ --- 4. **Comparison with stored answer** Stored correct answer: $D$ My derived answer is $A$. They do **not** match. The likely error in the stored answer is in the Reason statement: $$\vec{AB}+\vec{BC}+\vec{CD}+\vec{AD}=\vec{AD}+\vec{AD}=2\vec{AD}=4\vec{AO}$$ not $2\vec{AO}$. So Reason is false, and option $D$ cannot be correct.

View written solutionFree

Correct answer: A

We need to check whether $\vec{AB}+\vec{AC}+\vec{AD}=4\vec{AO}+\vec{OB}+\vec{OC}$ is true.

Using position-vector type decomposition: $\vec{AB}=\vec{AO}+\vec{OB}$ $\vec{AC}=\vec{AO}+\vec{OC}$ $\vec{AD}=\vec{AO}+\vec{OD}$ Hence,

=3\vec{AO}+\vec{OB}+\vec{OC}+\vec{OD}$$ So the assertion will be true if $$3\vec{AO}+\vec{OB}+\vec{OC}+\vec{OD}=4\vec{AO}+\vec{OB}+\vec{OC}$$ which requires $$\vec{OD}=\vec{AO}$$ Now, since $A,B,C,D$ are four points equally spaced on a semicircular arc, the arc is divided into 3 equal parts, so the central angle between consecutive points is $$\frac{180^\circ}{3}=60^\circ$$ Thus the points are at angles $180^\circ,120^\circ,60^\circ,0^\circ$ (for a suitable orientation), so $A$ and $D$ are endpoints of the diameter. Therefore $$\vec{OD}=-\vec{OA}=\vec{AO}$$ Hence the assertion is **correct**. --- 2. **Check the reason** The reason states: $$\vec{AB}+\vec{BC}+\vec{CD}+\vec{AD}=2\vec{AO}$$ Now, $$\vec{AB}+\vec{BC}+\vec{CD}=\vec{AD}$$ by polygon law. Therefore, $$\vec{AB}+\vec{BC}+\vec{CD}+\vec{AD}=2\vec{AD}$$ Since $A$ and $D$ are endpoints of a diameter and $O$ is the midpoint, $$\vec{AD}=2\vec{AO}$$ Therefore, $$2\vec{AD}=4\vec{AO}$$ not $2\vec{AO}$. So the stated reason is **incorrect**. --- 3. **Conclusion** - Assertion A is correct. - Reason R is not correct. Therefore the correct option is: $$\boxed{A}$$ --- 4. **Comparison with stored answer** Stored correct answer: $D$ My derived answer is $A$. They do **not** match. The likely error in the stored answer is in the Reason statement: $$\vec{AB}+\vec{BC}+\vec{CD}+\vec{AD}=\vec{AD}+\vec{AD}=2\vec{AD}=4\vec{AO}$$ not $2\vec{AO}$. So Reason is false, and option $D$ cannot be correct.