Question

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 $\vec{AB} + \vec{BC} + \vec{CD} = \vec{AD} = 2\vec{AO}$ 

• A. Both A and R are correct and R is the correct explanation of A.
• B. Both A and R are correct but R is not the correct explanation of A.
• C. A is correct but R is not correct.
• D. A is not correct but R is correct.

Hint:

In assertion-reason questions, first check if each statement is independently true. Then, check if the reason logically and completely explains the assertion. A correct reason must use the premises given in the assertion to derive the conclusion.

Answer 1

The Correct Option is B

To solve this problem, we need to analyze both the assertion and the reason in the context of vector addition.

1. Understanding the Assertion (A):
   The assertion states that if |\vec{AB}| = |\vec{BC}| = |\vec{CD}|\ for the points A, B, C, D on a semicircular arc with center at O, then \vec{AB} + \vec{AC} + \vec{AD} = 4\vec{AO} + \vec{OB} + \vec{OC}.
   • The equation involves using vector representations of line segments joining these points and applying properties of vectors in the semicircular geometry.
   • The expression 4\vec{AO} and adding vectors \vec{OB} + \vec{OC} seems to be derived from positioning and combining the vectors emanating from the center O.
2. Understanding the Reason (R):
   The reason claims the polygon law of vector addition gives \vec{AB} + \vec{BC} + \vec{CD} = \vec{AD} = 2\vec{AO}.
   • Based on the figure, using the polygon (or triangle) law for ABCD on the semicircular path implies a relation among these vectors.
   • The statement \vec{AD} = 2\vec{AO} seems incorrect because, geometrically, \vec{AD} should equal 2\vec{\text{radius}} (if specific symmetric positions were considered).
3. Conclusion:
   The assertion and reason are both correct in certain contexts:
   • The assertion (A) can hold true under specific arrangements where vectors are placed symmetrically.
   • The reason (R) involves a correct principle but does not correctly explain the given assertion's derivation.
   • The correct choice is: Both A and R are correct but R is not the correct explanation of A.

Diagram of points A, B, C, and D on a semi-circle.