If `|veca| =3, |vecb|=4 and |vecc|=5` such that each is perpendicular to sum of the other two, find `|veca + vecb+vecc|`

Answer

To solve the problem, we need to find the magnitude of the vector sum \(|\vec{a} + \vec{b} + \vec{c}|\) given that \(|\vec{a}| = 3\), \(|\vec{b}| = 4\), and \(|\vec{c}| = 5\), and that each vector is perpendicular to the sum of the other two.

### Step-by-Step Solution:

1. **Understanding the Perpendicular Condition:**
Since each vector is perpendicular to the sum of the other two, we can express this mathematically:
- \(\vec{a} \cdot (\vec{b} + \vec{c}) = 0\)
- \(\vec{b} \cdot (\vec{a} + \vec{c}) = 0\)
- \(\vec{c} \cdot (\vec{a} + \vec{b}) = 0\)

This implies that all three vectors are mutually perpendicular.

2. **Using the Dot Product:**
The dot product of two vectors is zero when they are perpendicular. Therefore, we can expand the dot products:
- \(\vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0\)
- \(\vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0\)
- \(\vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0\)

3. **Adding the Equations:**
Adding all three equations gives:
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]
This implies:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 0 \]

4. **Finding the Magnitude of the Sum:**
We want to find \(|\vec{a} + \vec{b} + \vec{c}|\). Using the property of dot products:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) \]
Expanding this gives:
\[ |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \]
Since \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 0\), we have:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 \]

5. **Substituting the Magnitudes:**
Now substituting the given magnitudes:
\[ |\vec{a}|^2 = 3^2 = 9 \]
\[ |\vec{b}|^2 = 4^2 = 16 \]
\[ |\vec{c}|^2 = 5^2 = 25 \]
Therefore:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 9 + 16 + 25 = 50 \]

6. **Calculating the Magnitude:**
Taking the square root gives:
\[ |\vec{a} + \vec{b} + \vec{c}| = \sqrt{50} = 5\sqrt{2} \]

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