Let ` vec a , vec b` and ` vec c`be three vectors such that `| vec a|=3,| vec b|=4,| vec c|=5`and each one of them being perpendicular to the sum of the other two, find `| vec a+ vec b+ vec c|`.

Options

A. `sqrt2`
B. `2sqrt2`
C. `5sqrt2`
D. `3sqrt2`

Explanation

To solve the problem, we are given three vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) with magnitudes \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \), and \( |\vec{c}| = 5 \). Each vector is perpendicular to the sum of the other two vectors. We need to find the magnitude of the vector sum \( |\vec{a} + \vec{b} + \vec{c}| \).

### Step-by-Step Solution:

1. **Understanding the Perpendicularity Condition:**
Since \( \vec{a} \) is perpendicular to \( \vec{b} + \vec{c} \), we can write:
\[ \vec{a} \cdot (\vec{b} + \vec{c}) = 0 \]
This implies:
\[ \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \quad \text{(1)} \]

2. **For the vector \( \vec{b} \):**
Similarly, since \( \vec{b} \) is perpendicular to \( \vec{a} + \vec{c} \):
\[ \vec{b} \cdot (\vec{a} + \vec{c}) = 0 \]
This implies:
\[ \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0 \quad \text{(2)} \]

3. **For the vector \( \vec{c} \):**
Lastly, since \( \vec{c} \) is perpendicular to \( \vec{a} + \vec{b} \):
\[ \vec{c} \cdot (\vec{a} + \vec{b}) = 0 \]
This implies:
\[ \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \quad \text{(3)} \]

4. **Using the Magnitudes:**
From the magnitudes given, we have:
\[ |\vec{a}|^2 = 3^2 = 9, \quad |\vec{b}|^2 = 4^2 = 16, \quad |\vec{c}|^2 = 5^2 = 25 \]

5. **Calculating the Magnitude of the Sum:**
We can find \( |\vec{a} + \vec{b} + \vec{c}|^2 \) 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 from equations (1), (2), and (3), we know that:
\[ \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0, \quad \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0, \quad \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \]
This means:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 0 \]

6. **Substituting into the Magnitude Formula:**
Thus, we have:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 9 + 16 + 25 + 2(0) = 50 \]

7. **Finding the Final Magnitude:**
Therefore:
\[ |\vec{a} + \vec{b} + \vec{c}| = \sqrt{50} = 5\sqrt{2} \]

### Final Answer:
\[ |\vec{a} + \vec{b} + \vec{c}| = 5\sqrt{2} \]