Let `vecA , vecB and vecC` be vectors of legth , 3,4and 5 respectively. Let `vecA` be perpendicular to `vecB + vecC, vecB " to " vecC + vecA and vecC " to" vecA + vecB` then the length of vector `vecA + vecB+ vecC` is __________.

Explanation

To solve the problem, we need to find the length of the vector \(\vec{A} + \vec{B} + \vec{C}\) given the conditions about the perpendicularity of the vectors.

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
- We know that:
- \(\vec{A} \perp (\vec{B} + \vec{C})\)
- \(\vec{B} \perp (\vec{C} + \vec{A})\)
- \(\vec{C} \perp (\vec{A} + \vec{B})\)

2. **Using the Dot Product**:
- From the property of perpendicular vectors, we can write:
- \(\vec{A} \cdot (\vec{B} + \vec{C}) = 0\) (1)
- \(\vec{B} \cdot (\vec{C} + \vec{A}) = 0\) (2)
- \(\vec{C} \cdot (\vec{A} + \vec{B}) = 0\) (3)

3. **Expanding the Dot Products**:
- Expanding each equation gives:
- From (1): \(\vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C} = 0\) (4)
- From (2): \(\vec{B} \cdot \vec{C} + \vec{B} \cdot \vec{A} = 0\) (5)
- From (3): \(\vec{C} \cdot \vec{A} + \vec{C} \cdot \vec{B} = 0\) (6)

4. **Adding the Equations**:
- Adding equations (4), (5), and (6):
\[ (\vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}) + (\vec{B} \cdot \vec{C} + \vec{B} \cdot \vec{A}) + (\vec{C} \cdot \vec{A} + \vec{C} \cdot \vec{B}) = 0 \]
- This simplifies to:
\[ 2(\vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{C} + \vec{C} \cdot \vec{A}) = 0 \]
- Therefore, we conclude:
\[ \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{C} + \vec{C} \cdot \vec{A} = 0 \quad (7) \]

5. **Calculating the Length of \(\vec{A} + \vec{B} + \vec{C}\)**:
- The length (magnitude) of \(\vec{A} + \vec{B} + \vec{C}\) is given by:
\[ |\vec{A} + \vec{B} + \vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + |\vec{C}|^2 + 2(\vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{C} + \vec{C} \cdot \vec{A}) \]
- Substituting the lengths:
- \(|\vec{A}| = 3\), \(|\vec{B}| = 4\), \(|\vec{C}| = 5\)
- Therefore:
\[ |\vec{A} + \vec{B} + \vec{C}|^2 = 3^2 + 4^2 + 5^2 + 2 \cdot 0 \]
\[ = 9 + 16 + 25 = 50 \]

6. **Finding the Magnitude**:
- Taking the square root gives:
\[ |\vec{A} + \vec{B} + \vec{C}| = \sqrt{50} = 5\sqrt{2} \]

### Final Answer:
The length of the vector \(\vec{A} + \vec{B} + \vec{C}\) is \(5\sqrt{2}\).