If `|A|=3,|B|=4,|C|=5" and "a,b,c` are such that each is perpendicular to the sum of other two, then `|a+b+c|` is

Options

A. `5sqrt(2)`
B. `(5)/(sqrt(2))`
C. `10sqrt(2)`
D. `10sqrt(3)`

Explanation

To solve the problem, we are given the magnitudes of three vectors \( |A| = 3 \), \( |B| = 4 \), and \( |C| = 5 \). We also know that each vector is perpendicular to the sum of the other two. We need to find the magnitude of the vector sum \( |A + B + C| \).

### Step-by-Step Solution:

1. **Understanding Perpendicularity**:
Since \( A \) is perpendicular to \( B + C \), we have:
\[ A \cdot (B + C) = 0 \implies A \cdot B + A \cdot C = 0 \]
Similarly, we can write:
\[ B \cdot (A + C) = 0 \implies B \cdot A + B \cdot C = 0 \]
\[ C \cdot (A + B) = 0 \implies C \cdot A + C \cdot B = 0 \]

2. **Using the Dot Product**:
From the above equations, we can conclude:
\[ A \cdot B + A \cdot C = 0 \quad (1) \]
\[ B \cdot A + B \cdot C = 0 \quad (2) \]
\[ C \cdot A + C \cdot B = 0 \quad (3) \]
Adding equations (1), (2), and (3):
\[ (A \cdot B + A \cdot C) + (B \cdot A + B \cdot C) + (C \cdot A + C \cdot B) = 0 \]
This simplifies to:
\[ 2(A \cdot B + B \cdot C + C \cdot A) = 0 \implies A \cdot B + B \cdot C + C \cdot A = 0 \]

3. **Finding the Magnitude of the Sum**:
We will now use the formula for the magnitude of the sum of vectors:
\[ |A + B + C|^2 = |A|^2 + |B|^2 + |C|^2 + 2(A \cdot B + B \cdot C + C \cdot A) \]
Since \( A \cdot B + B \cdot C + C \cdot A = 0 \), we can simplify this to:
\[ |A + B + C|^2 = |A|^2 + |B|^2 + |C|^2 \]

4. **Calculating the Magnitudes**:
Now substituting the values:
\[ |A|^2 = 3^2 = 9, \quad |B|^2 = 4^2 = 16, \quad |C|^2 = 5^2 = 25 \]
Therefore:
\[ |A + B + C|^2 = 9 + 16 + 25 = 50 \]

5. **Taking the Square Root**:
Finally, we take the square root to find the magnitude:
\[ |A + B + C| = \sqrt{50} = 5\sqrt{2} \]

### Final Answer:
\[ |A + B + C| = 5\sqrt{2} \]