If `vecA , vecB` and `vecC` are mutually perpendiculasr (no zero vectors) to each other, then choose the correct option.

Options

A. `(vecA+vecB).vecC=AC+BC`
B. `vecA.(vecBxxvecC)=0`
C. `(vecAxxvecB).vecC = 0`
D. `vecAxx(vecBxxvecC)= 0`

Explanation

To solve the problem, we need to analyze the properties of mutually perpendicular vectors. Let's denote the vectors as \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\). Since these vectors are mutually perpendicular, we can use the properties of dot and cross products.

### Step-by-step Solution:

1. **Understanding the Vectors**:
- Given that \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) are mutually perpendicular, we can visualize them as the x, y, and z axes in a 3D coordinate system.
- This means:
\[ \vec{A} \cdot \vec{B} = 0, \quad \vec{B} \cdot \vec{C} = 0, \quad \vec{C} \cdot \vec{A} = 0 \]

2. **Evaluating the Options**:
- We need to evaluate the given options to find the correct one.

3. **Option A: \(\vec{A} + \vec{B}\)**:
- The magnitude of \(\vec{A} + \vec{B}\) is given by:
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2} \]
- When we dot this with \(\vec{C}\):
\[ (\vec{A} + \vec{B}) \cdot \vec{C} = \vec{A} \cdot \vec{C} + \vec{B} \cdot \vec{C} = 0 + 0 = 0 \]
- This option is incorrect as it does not yield a non-zero result.

4. **Option B: \(\vec{A} \cdot (\vec{B} \times \vec{C})\)**:
- The cross product \(\vec{B} \times \vec{C}\) gives a vector that is perpendicular to both \(\vec{B}\) and \(\vec{C}\), which is in the direction of \(\vec{A}\).
- Therefore:
\[ \vec{A} \cdot (\vec{B} \times \vec{C}) = |\vec{A}| |\vec{B}| |\vec{C}| \sin(90^\circ) = |\vec{A}| |\vec{B}| |\vec{C}| \]
- This option is correct as it yields a non-zero result.

5. **Option C: \(\vec{B} \cdot (\vec{C} \times \vec{A})\)**:
- Similarly, \(\vec{C} \times \vec{A}\) gives a vector in the direction of \(\vec{B}\).
- Thus:
\[ \vec{B} \cdot (\vec{C} \times \vec{A}) = |\vec{B}| |\vec{C}| |\vec{A}| \sin(90^\circ) = |\vec{B}| |\vec{C}| |\vec{A}| \]
- This option is also correct.

6. **Option D: \(\vec{B} \times \vec{C}\)**:
- The cross product \(\vec{B} \times \vec{C}\) is perpendicular to both \(\vec{B}\) and \(\vec{C}\), which means it is in the direction of \(\vec{A}\).
- If we take \(\vec{A} \times (\vec{B} \times \vec{C})\), we can use the vector triple product identity:
\[ \vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \vec{B} - (\vec{A} \cdot \vec{B}) \vec{C} \]
- Since \(\vec{A} \cdot \vec{B} = 0\) and \(\vec{A} \cdot \vec{C} = 0\), this results in:
\[ \vec{A} \times (\vec{B} \times \vec{C}) = 0 \]
- This option is incorrect.

### Conclusion:
The correct options are B and C, as they yield non-zero results. However, the question asks for the correct option, and since only one option can be chosen, we conclude that:

**Correct Option: B or C (depending on the context of the question).**