To solve this problem, we need to identify which vector is perpendicular to the plane defined by the given vectors. In this context, the plane is defined by the two vectors, and the perpendicular vector to this plane can be found using the cross product of these vectors.

Given:

1. Vectors defining the plane:
   • Vector A: $\mathbf{A} = 3\mathbf{m} - 7\mathbf{m}$
   • Vector B: $\mathbf{B} = 2\mathbf{m}$

Let's assume the vectors $\mathbf{A}$ and $\mathbf{B}$ are:

• $\mathbf{A} = (3m - 2m) \mathbf{i} = m \mathbf{i}$
• $\mathbf{B} = 2m \mathbf{j}$

Cross Product:

The cross product of $\mathbf{A}$ and $\mathbf{B}$ gives us a vector perpendicular to both $\mathbf{A}$ and $\mathbf{B}$ and hence perpendicular to the plane:

$\mathbf{C} = \mathbf{A} \times \mathbf{B} = m \mathbf{i} \times 2m \mathbf{j}$

$\mathbf{C} = 2m^2 \mathbf{k}$

Conclusion:

The vector $\mathbf{C} = 2m^2 \mathbf{k}$ is perpendicular to the plane defined by the two given vectors.

Thus, the correct vector that is perpendicular to the plane is $\mathbf{k}$ or any scalar multiple of it.

---

Would you like more details on any part of the solution? Here are some related questions to consider:

1. How do you compute the cross product of two vectors in 3D space?
2. What is the significance of the cross product in vector analysis?
3. How can we determine the direction of a vector resulting from a cross product?
4. What is the geometric interpretation of the cross product?
5. How do you determine if two vectors are parallel or perpendicular using their cross product?

Tip: The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.