`vecA, vecB" and "vecC` are three orthogonal vectors with magnitudes 3, 4 and 12 respectively. The value of `|vecA-vecB+vecC|` will be :-

Options

A. 11
B. 19
C. 13
D. can't be determined

Answer

To solve the problem, we need to find the magnitude of the vector expression \(|\vec{A} - \vec{B} + \vec{C}|\) given that \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) are orthogonal vectors with magnitudes 3, 4, and 12 respectively.

### Step-by-Step Solution:

1. **Understanding Orthogonal Vectors**:
- Orthogonal vectors are vectors that are perpendicular to each other. This means that the dot product of any two different vectors among \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) is zero.
- Therefore, we have:
\[ \vec{A} \cdot \vec{B} = 0, \quad \vec{B} \cdot \vec{C} = 0, \quad \vec{C} \cdot \vec{A} = 0 \]

2. **Magnitude of Each Vector**:
- The magnitudes of the vectors are given as:
\[ |\vec{A}| = 3, \quad |\vec{B}| = 4, \quad |\vec{C}| = 12 \]

3. **Using the Magnitude Formula**:
- To find the magnitude of the vector \(\vec{A} - \vec{B} + \vec{C}\), we will use the formula for the magnitude of a vector:
\[ |\vec{X}| = \sqrt{\vec{X} \cdot \vec{X}} \]
- Here, \(\vec{X} = \vec{A} - \vec{B} + \vec{C}\).

4. **Expanding the Magnitude**:
- We can expand the expression \(|\vec{A} - \vec{B} + \vec{C}|^2\):
\[ |\vec{A} - \vec{B} + \vec{C}|^2 = (\vec{A} - \vec{B} + \vec{C}) \cdot (\vec{A} - \vec{B} + \vec{C}) \]
- This expands to:
\[ |\vec{A}|^2 + |\vec{B}|^2 + |\vec{C}|^2 - 2(\vec{A} \cdot \vec{B}) + 2(\vec{A} \cdot \vec{C}) - 2(\vec{B} \cdot \vec{C}) \]

5. **Substituting Known Values**:
- Since \(\vec{A} \cdot \vec{B} = 0\), \(\vec{A} \cdot \vec{C} = 0\), and \(\vec{B} \cdot \vec{C} = 0\), the equation simplifies to:
\[ |\vec{A} - \vec{B} + \vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + |\vec{C}|^2 \]

6. **Calculating the Squares of the Magnitudes**:
- Now, substituting the magnitudes:
\[ |\vec{A}|^2 = 3^2 = 9, \quad |\vec{B}|^2 = 4^2 = 16, \quad |\vec{C}|^2 = 12^2 = 144 \]
- Therefore:
\[ |\vec{A} - \vec{B} + \vec{C}|^2 = 9 + 16 + 144 = 169 \]

7. **Taking the Square Root**:
- Finally, we take the square root to find the magnitude:
\[ |\vec{A} - \vec{B} + \vec{C}| = \sqrt{169} = 13 \]

### Final Answer:
The value of \(|\vec{A} - \vec{B} + \vec{C}|\) is \(13\).