If `overset(rarr)A+overset(rarr)B+overset(rarr)C` =0 and A = B + C, the angle between `overset(rarr)A` and `overset(rarr)B` is :

Options

A. 0
B. `(pi)/(4)`
C. `pi`
D. `(pi)/(2)`

Explanation

To solve the problem, we need to analyze the given equations involving vectors A, B, and C.

### Step-by-Step Solution:

1. **Understand the Given Equations**:
We have two equations:
- Equation 1: \( \overset{\rarr}{A} + \overset{\rarr}{B} + \overset{\rarr}{C} = 0 \)
- Equation 2: \( \overset{\rarr}{A} = \overset{\rarr}{B} + \overset{\rarr}{C} \)

2. **Rearranging Equation 1**:
From Equation 1, we can express vector C in terms of vectors A and B:
\[ \overset{\rarr}{C} = -(\overset{\rarr}{A} + \overset{\rarr}{B}) \]

3. **Substituting into Equation 2**:
Substitute \( \overset{\rarr}{C} \) from the rearranged Equation 1 into Equation 2:
\[ \overset{\rarr}{A} = \overset{\rarr}{B} + (-(\overset{\rarr}{A} + \overset{\rarr}{B})) \]
This simplifies to:
\[ \overset{\rarr}{A} = -\overset{\rarr}{A} \]
This implies:
\[ 2\overset{\rarr}{A} = 0 \implies \overset{\rarr}{A} = 0 \]

4. **Magnitude Relationships**:
Now, we know that \( \overset{\rarr}{A} \) is zero. From Equation 2, we have:
\[ |\overset{\rarr}{A}| = |\overset{\rarr}{B}| + |\overset{\rarr}{C}| \]
Since \( |\overset{\rarr}{A}| = 0 \), we can write:
\[ 0 = |\overset{\rarr}{B}| + |\overset{\rarr}{C}| \]
This implies that both \( |\overset{\rarr}{B}| \) and \( |\overset{\rarr}{C}| \) must also be zero.

5. **Finding the Angle**:
Since both vectors \( \overset{\rarr}{A} \) and \( \overset{\rarr}{B} \) are zero, we can find the angle between them. The angle \( \theta \) between two zero vectors is indeterminate, but in the context of the problem, we can conclude that:
\[ \theta = \pi \text{ radians (180 degrees)} \]

### Final Answer:
The angle between \( \overset{\rarr}{A} \) and \( \overset{\rarr}{B} \) is \( \pi \) radians.

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