If `veca` an `vecb` are perpendicular vectors, `|veca+vecb|=13 and |veca|=5`, find the value of `|vecb|`.

Answer

To solve the problem, we need to find the magnitude of vector \(\vec{b}\) given that \(\vec{a}\) and \(\vec{b}\) are perpendicular vectors, \(|\vec{a} + \vec{b}| = 13\), and \(|\vec{a}| = 5\).

### Step-by-Step Solution:

1. **Understand the relationship between the vectors**:
Since \(\vec{a}\) and \(\vec{b}\) are perpendicular, we know that:
\[ \vec{a} \cdot \vec{b} = 0 \]

2. **Use the given magnitudes**:
We know:
\[ |\vec{a}| = 5 \quad \text{and} \quad |\vec{a} + \vec{b}| = 13 \]

3. **Apply the formula for the magnitude of the sum of two vectors**:
The magnitude of the sum of two vectors can be expressed as:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) \]
Since \(\vec{a} \cdot \vec{b} = 0\), this simplifies to:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \]

4. **Substituting the known values**:
We can substitute the known values into the equation:
\[ 13^2 = 5^2 + |\vec{b}|^2 \]
This gives us:
\[ 169 = 25 + |\vec{b}|^2 \]

5. **Isolate \(|\vec{b}|^2\)**:
Rearranging the equation to solve for \(|\vec{b}|^2\):
\[ |\vec{b}|^2 = 169 - 25 \]
\[ |\vec{b}|^2 = 144 \]

6. **Take the square root**:
To find \(|\vec{b}|\), take the square root of both sides:
\[ |\vec{b}| = \sqrt{144} = 12 \]

### Final Answer:
\[ |\vec{b}| = 12 \]