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 will follow these steps:

### Step 1: Write down the given information
We are given three vectors \( \vec{a}, \vec{b}, \vec{c} \) with the following magnitudes:
- \( |\vec{a}| = 3 \)
- \( |\vec{b}| = 4 \)
- \( |\vec{c}| = 5 \)

Additionally, each vector is perpendicular to the sum of the other two vectors.

### Step 2: Set up the perpendicular conditions
From the problem statement, we know:
1. \( \vec{a} \cdot (\vec{b} + \vec{c}) = 0 \)
2. \( \vec{b} \cdot (\vec{a} + \vec{c}) = 0 \)
3. \( \vec{c} \cdot (\vec{a} + \vec{b}) = 0 \)

This implies that:
- \( \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \)
- \( \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0 \)
- \( \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \)

### Step 3: Calculate the magnitude of \( |\vec{a} + \vec{b} + \vec{c}| \)
To find \( |\vec{a} + \vec{b} + \vec{c}| \), we use the formula:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) \]
Expanding this, we have:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{c} \cdot \vec{c} + 2(\vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c}) \]

### Step 4: Substitute known values
We know:
- \( \vec{a} \cdot \vec{a} = |\vec{a}|^2 = 3^2 = 9 \)
- \( \vec{b} \cdot \vec{b} = |\vec{b}|^2 = 4^2 = 16 \)
- \( \vec{c} \cdot \vec{c} = |\vec{c}|^2 = 5^2 = 25 \)

Now we substitute these values:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 9 + 16 + 25 + 2(\vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c}) \]
Since \( \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \) and \( \vec{b} \cdot \vec{c} + \vec{b} \cdot \vec{a} = 0 \) and \( \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \), we find that:
\[ \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c} = 0 \]
Thus, we can simplify:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 9 + 16 + 25 + 2(0) = 50 \]

### Step 5: Find the magnitude
Taking the square root gives us:
\[ |\vec{a} + \vec{b} + \vec{c}| = \sqrt{50} = 5\sqrt{2} \]

### Final Answer
Thus, the magnitude of \( \vec{a} + \vec{b} + \vec{c} \) is:
\[ |\vec{a} + \vec{b} + \vec{c}| = 5\sqrt{2} \]

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