Let a, b, c be unit vectors such that `a + b + c = 0` which one of the following is correct ?

Options

A. `a xx b = b xx c = c xx a = 0`
B. `a xx b = b xx c = c xx a ne 0`
C. `a xx b = b xx c = a xx c = 0`
D. `a xx b, b xx c, c xxa ` are mutually perpendicular

Answer

To solve the problem, we need to analyze the given condition that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \) where \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are unit vectors.

### Step-by-step Solution:

1. **Understanding the Condition**:
Since \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are unit vectors, we know that:
\[ |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = 1 \]
The equation \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \) implies that \( \mathbf{c} = -(\mathbf{a} + \mathbf{b}) \).

2. **Taking the Magnitude**:
Taking the magnitude of both sides:
\[ |\mathbf{c}| = |-(\mathbf{a} + \mathbf{b})| = |\mathbf{a} + \mathbf{b}| \]
Since \( |\mathbf{c}| = 1 \):
\[ 1 = |\mathbf{a} + \mathbf{b}| \]

3. **Using the Magnitude Formula**:
We can use the formula for the magnitude of the sum of two vectors:
\[ |\mathbf{a} + \mathbf{b}| = \sqrt{|\mathbf{a}|^2 + |\mathbf{b}|^2 + 2(\mathbf{a} \cdot \mathbf{b})} \]
Substituting \( |\mathbf{a}| = 1 \) and \( |\mathbf{b}| = 1 \):
\[ 1 = \sqrt{1^2 + 1^2 + 2(\mathbf{a} \cdot \mathbf{b})} \]
This simplifies to:
\[ 1 = \sqrt{2 + 2(\mathbf{a} \cdot \mathbf{b})} \]

4. **Squaring Both Sides**:
Squaring both sides gives:
\[ 1 = 2 + 2(\mathbf{a} \cdot \mathbf{b}) \]
Rearranging this:
\[ 2(\mathbf{a} \cdot \mathbf{b}) = 1 - 2 \]
\[ 2(\mathbf{a} \cdot \mathbf{b}) = -1 \]
\[ \mathbf{a} \cdot \mathbf{b} = -\frac{1}{2} \]

5. **Conclusion**:
The dot product \( \mathbf{a} \cdot \mathbf{b} = -\frac{1}{2} \) indicates that the angle \( \theta \) between \( \mathbf{a} \) and \( \mathbf{b} \) is:
\[ \cos \theta = -\frac{1}{2} \implies \theta = 120^\circ \]
Therefore, the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are mutually inclined at \( 120^\circ \) to each other.

### Final Answer:
The correct option is that \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are mutually inclined at \( 120^\circ \).