Assertion: If I is the incentre of `/_\ABC, then`|vec(BC)| vec(IA) +|vec(CA)| vec(IB) +|vec(AB)| vec(IC) =0` Reason: If O is the origin, then the position vector of centroid of `/_\ABC` is (vec(OA)+vec(OB)+vec(OC))/3`

Options

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 incorrect
D. R is correct but A is incorrect

Explanation

To solve the given problem, we need to analyze the assertion and the reason step by step.

### Step 1: Understand the Assertion
The assertion states that if I is the incenter of triangle ABC, then:
\[ |\vec{BC}| \vec{IA} + |\vec{CA}| \vec{IB} + |\vec{AB}| \vec{IC} = 0 \]

### Step 2: Define the Vectors
Let:
- \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \) be the position vectors of points A, B, and C respectively.
- \( \vec{I} \) be the position vector of the incenter I.

### Step 3: Express the Vectors in Terms of Position Vectors
The vectors \( \vec{IA} \), \( \vec{IB} \), and \( \vec{IC} \) can be expressed as:
- \( \vec{IA} = \vec{A} - \vec{I} \)
- \( \vec{IB} = \vec{B} - \vec{I} \)
- \( \vec{IC} = \vec{C} - \vec{I} \)

### Step 4: Substitute the Vectors into the Assertion
Substituting these expressions into the assertion gives:
\[ |\vec{BC}| (\vec{A} - \vec{I}) + |\vec{CA}| (\vec{B} - \vec{I}) + |\vec{AB}| (\vec{C} - \vec{I}) = 0 \]

### Step 5: Expand the Equation
Expanding this equation results in:
\[ |\vec{BC}| \vec{A} - |\vec{BC}| \vec{I} + |\vec{CA}| \vec{B} - |\vec{CA}| \vec{I} + |\vec{AB}| \vec{C} - |\vec{AB}| \vec{I} = 0 \]

### Step 6: Combine Like Terms
Combining like terms gives:
\[ (|\vec{BC}| \vec{A} + |\vec{CA}| \vec{B} + |\vec{AB}| \vec{C}) - (|\vec{BC}| + |\vec{CA}| + |\vec{AB}|) \vec{I} = 0 \]

### Step 7: Rearranging the Equation
Rearranging this leads to:
\[ |\vec{BC}| \vec{A} + |\vec{CA}| \vec{B} + |\vec{AB}| \vec{C} = (|\vec{BC}| + |\vec{CA}| + |\vec{AB}|) \vec{I} \]

### Step 8: Conclusion for Assertion
This shows that the assertion is true because the weighted sum of the position vectors equals the weighted sum of the incenter's position vector.

### Step 9: Analyze the Reason
The reason states that if O is the origin, then the position vector of the centroid of triangle ABC is given by:
\[ \frac{\vec{OA} + \vec{OB} + \vec{OC}}{3} \]

### Step 10: Verify the Reason
The position vector of the centroid G of triangle ABC can be expressed as:
\[ \vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3} \]
This confirms that the reason is also correct.

### Final Conclusion
Both the assertion and the reason are true, but the reason does not explain the assertion correctly. Therefore, the correct option is that both are true, but the reason is not the correct explanation for the assertion.