Let `2veca = vecb xx vecc + 2vecb` where `veca, vecb and vecc` are three unit vectors, then sum of all possible values of `|3veca + 4vecb + 5vecc|` is

Options

A. 10
B. 12
C. 14
D. 16

Answer

To solve the problem, we need to analyze the given equation and find the sum of all possible values of the expression \( |3\vec{a} + 4\vec{b} + 5\vec{c}| \).

### Step-by-step Solution:

1. **Given Equation**:
We start with the equation:
\[ 2\vec{a} = \vec{b} \times \vec{c} + 2\vec{b} \]

2. **Dot Product with \(\vec{b}\)**:
We take the dot product of both sides with \(\vec{b}\):
\[ 2\vec{a} \cdot \vec{b} = (\vec{b} \times \vec{c}) \cdot \vec{b} + 2\vec{b} \cdot \vec{b} \]
Since \((\vec{b} \times \vec{c})\) is perpendicular to \(\vec{b}\), the dot product \((\vec{b} \times \vec{c}) \cdot \vec{b} = 0\). Thus, we have:
\[ 2\vec{a} \cdot \vec{b} = 2|\vec{b}|^2 \]

3. **Magnitude of Unit Vectors**:
Since \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are unit vectors, we know that \(|\vec{b}| = 1\). Therefore:
\[ 2\vec{a} \cdot \vec{b} = 2 \implies \vec{a} \cdot \vec{b} = 1 \]
This implies that \(\vec{a}\) and \(\vec{b}\) are parallel and since they are unit vectors, we have:
\[ \vec{a} = \vec{b} \]

4. **Substituting \(\vec{b}\)**:
Substitute \(\vec{b}\) in the original equation:
\[ 2\vec{a} = \vec{a} \times \vec{c} + 2\vec{a} \]
This simplifies to:
\[ \vec{a} \times \vec{c} = 0 \]
This means \(\vec{a}\) and \(\vec{c}\) are also parallel.

5. **Finding Relationships**:
Since \(\vec{a}\) and \(\vec{c}\) are parallel, we have two cases:
- Case 1: \(\vec{c} = \vec{a}\)
- Case 2: \(\vec{c} = -\vec{a}\)

6. **Calculating \( |3\vec{a} + 4\vec{b} + 5\vec{c}| \)**:
- **For Case 1** (\(\vec{c} = \vec{a}\)):
\[ |3\vec{a} + 4\vec{a} + 5\vec{a}| = |(3 + 4 + 5)\vec{a}| = |12\vec{a}| = 12 \]

- **For Case 2** (\(\vec{c} = -\vec{a}\)):
\[ |3\vec{a} + 4\vec{a} + 5(-\vec{a})| = |(3 + 4 - 5)\vec{a}| = |2\vec{a}| = 2 \]

7. **Sum of All Possible Values**:
The possible values of \( |3\vec{a} + 4\vec{b} + 5\vec{c}| \) are \(12\) and \(2\). Therefore, the sum of all possible values is:
\[ 12 + 2 = 14 \]

### Final Answer:
The sum of all possible values of \( |3\vec{a} + 4\vec{b} + 5\vec{c}| \) is \( \boxed{14} \).