If `vec(a).vec(b) and vec(c )` are unit vectors such that `vec(a)` is perpendicular to the plane of `vec(b) , vec(c )` and the angle between `vec(b) and vec(c )` is `(pi)/(3)`. Then what is `|vec(a) +vec(b)+vec(c )|` ?

Options

A. A) 1
B. B) 2
C. C) 3
D. D) 4

Answer

To solve the problem, we need to find the magnitude of the vector sum \( \vec{a} + \vec{b} + \vec{c} \) given that \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are unit vectors, \( \vec{a} \) is perpendicular to the plane formed by \( \vec{b} \) and \( \vec{c} \), and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{3} \).

### Step-by-step Solution:

1. **Identify the properties of the vectors**:
- Since \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are unit vectors, we have:
\[ |\vec{a}| = 1, \quad |\vec{b}| = 1, \quad |\vec{c}| = 1 \]

2. **Calculate the dot product \( \vec{b} \cdot \vec{c} \)**:
- The angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{3} \). Therefore, using the dot product formula:
\[ \vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos\left(\frac{\pi}{3}\right) = 1 \cdot 1 \cdot \frac{1}{2} = \frac{1}{2} \]

3. **Use the formula for the magnitude of the sum of vectors**:
- The magnitude of the sum of vectors can be calculated using the formula:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \]
- Since \( \vec{a} \) is perpendicular to both \( \vec{b} \) and \( \vec{c} \), we have:
\[ \vec{a} \cdot \vec{b} = 0 \quad \text{and} \quad \vec{c} \cdot \vec{a} = 0 \]

4. **Substituting the values into the formula**:
- Now substituting the known values:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 1^2 + 1^2 + 1^2 + 2(0 + \frac{1}{2} + 0) \]
- Simplifying this gives:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 1 + 1 + 1 + 2 \cdot \frac{1}{2} = 3 + 1 = 4 \]

5. **Taking the square root**:
- Finally, taking the square root to find the magnitude:
\[ |\vec{a} + \vec{b} + \vec{c}| = \sqrt{4} = 2 \]

### Final Answer:
\[ |\vec{a} + \vec{b} + \vec{c}| = 2 \]