Which of the following relations is correct?

Options

A. `vecA+B=vecC`
B. `vecA+vecB=C`
C. `vecA+vecB=vecC`
D. `A+vecB=vecC`

Answer

To solve the question, we need to analyze each of the given options to determine which relation is correct based on the properties of vectors and magnitudes.

### Step-by-Step Solution:

1. **Understanding Vectors and Magnitudes**:
- Vectors are quantities that have both magnitude and direction (e.g., velocity, force).
- The magnitude of a vector is a scalar quantity that represents the size or length of the vector (e.g., speed).

2. **Analyzing the Options**:
- **Option A**: \( \mathbf{a} + |\mathbf{b}| = \mathbf{c} \)
- Here, we are adding a vector \( \mathbf{a} \) and the magnitude of vector \( \mathbf{b} \) (which is a scalar). This operation is not valid because you cannot add a vector and a scalar directly. Hence, this option is incorrect.

- **Option B**: \( \mathbf{a} + \mathbf{b} = |\mathbf{c}| \)
- This implies that the sum of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) equals the magnitude of vector \( \mathbf{c} \). This is not a valid vector equation because the left side is a vector while the right side is a scalar. Hence, this option is incorrect.

- **Option C**: \( \mathbf{a} + \mathbf{b} = \mathbf{c} \)
- This states that the sum of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) results in another vector \( \mathbf{c} \). This is a valid vector equation, as vectors can be added together to yield another vector. Hence, this option is correct.

- **Option D**: \( |\mathbf{a}| + \mathbf{b} = \mathbf{c} \)
- Here, we are adding the magnitude of vector \( \mathbf{a} \) (a scalar) to vector \( \mathbf{b} \). This operation is not valid because you cannot add a scalar and a vector directly. Hence, this option is incorrect.

3. **Conclusion**:
- After analyzing all the options, we find that the only correct relation is given by **Option C**: \( \mathbf{a} + \mathbf{b} = \mathbf{c} \).

### Final Answer:
**The correct relation is Option C: \( \mathbf{a} + \mathbf{b} = \mathbf{c} \)**.