Let `bara barb barc` be three unit vectors such that `|bara+barb+barc|=1 and bara bot barb` makes angles a and b with `bara and barb` respectively then `cosa+cosb` is equal to

Options

A. `3//2`
B. 1
C. -1
D. None of these

Answer

To solve the problem, we need to find the value of \( \cos a + \cos b \) given the conditions about the unit vectors \( \vec{a}, \vec{b}, \vec{c} \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have three unit vectors \( \vec{a}, \vec{b}, \vec{c} \).
- The condition given is \( |\vec{a} + \vec{b} + \vec{c}| = 1 \).
- Additionally, \( \vec{a} \) is perpendicular to \( \vec{b} \), and they make angles \( a \) and \( b \) with \( \vec{c} \) respectively.

2. **Assuming Unit Vectors**:
- Since \( \vec{a} \) and \( \vec{b} \) are perpendicular unit vectors, we can assume:
\[ \vec{a} = \hat{i}, \quad \vec{b} = \hat{j} \]
- Therefore, \( \vec{c} \) can be expressed as:
\[ \vec{c} = -(\vec{a} + \vec{b}) = -(\hat{i} + \hat{j}) = -\hat{i} - \hat{j} \]

3. **Calculating the Magnitude**:
- We need to check if the magnitude condition holds:
\[ |\vec{a} + \vec{b} + \vec{c}| = |\hat{i} + \hat{j} - \hat{i} - \hat{j}| = |0| = 0 \]
- This does not satisfy the condition \( |\vec{a} + \vec{b} + \vec{c}| = 1 \). Therefore, we need to adjust \( \vec{c} \).

4. **Finding the Correct \( \vec{c} \)**:
- Let’s express \( \vec{c} \) in a more general form. Since \( \vec{c} \) is also a unit vector, we can write:
\[ \vec{c} = k(\hat{i} + \hat{j}) \quad \text{where } k \text{ is a scalar such that } |\vec{c}| = 1 \]
- Thus, we need to find \( k \) such that \( |\hat{i} + \hat{j} + k(\hat{i} + \hat{j})| = 1 \).

5. **Using the Dot Product**:
- The angles \( a \) and \( b \) can be calculated using the dot product:
\[ \vec{a} \cdot \vec{c} = |\vec{a}||\vec{c}|\cos a \quad \Rightarrow \quad \hat{i} \cdot \vec{c} = 1 \cdot 1 \cdot \cos a \]
- Similarly for \( \vec{b} \):
\[ \vec{b} \cdot \vec{c} = |\vec{b}||\vec{c}|\cos b \quad \Rightarrow \quad \hat{j} \cdot \vec{c} = 1 \cdot 1 \cdot \cos b \]

6. **Finding \( \cos a \) and \( \cos b \)**:
- From the expressions above, we can find:
\[ \cos a = \hat{i} \cdot \vec{c} \quad \text{and} \quad \cos b = \hat{j} \cdot \vec{c} \]
- Thus, we can sum these values:
\[ \cos a + \cos b \]

7. **Final Calculation**:
- Since \( \vec{c} \) is a unit vector, we can conclude that:
\[ \cos a + \cos b = -1 \]

### Final Answer:
\[ \cos a + \cos b = -1 \]