Looking at the diagram of the parallelepiped $ABCDA_1B_1C_1D_1$, the points $M$ and $N$ are midpoints of edges $CC_1$ and $BC$ respectively.

Now, let's analyze the given statements:

1. $\vec{BC_1} \parallel \vec{CB_1}$:

   • This is true because the vector from $B$ to $C_1$ and the vector from $C$ to $B_1$ are parallel due to the geometric symmetry of the parallelepiped. These vectors represent opposite edges of the same plane.

2. $\vec{NM} \parallel \vec{BC_1}$:

   • This is false. The vector $\vec{NM}$ (from $N$ to $M$) does not point in the same direction as the vector from $B$ to $C_1$.

3. $\vec{BC} = \vec{A_1D_1}$:

   • This is false because $\vec{BC}$ and $\vec{A_1D_1}$ are not equal. They are parallel but do not have the same magnitude and direction.

4. $\vec{MN} = \vec{AD_1}$:

   • This is true. Since $M$ and $N$ are midpoints and due to the symmetry of the parallelepiped, $\vec{MN}$ is parallel and equal in length to $\vec{AD_1}$.

The correct answer is:

• $\vec{MN} = \vec{AD_1}$.

Would you like a further breakdown or more details on any of these statements?

Here are some related questions:

1. What is the geometric significance of a parallelepiped?
2. How do midpoints of edges in a parallelepiped affect the vector relations?
3. Can two vectors of different magnitude still be parallel?
4. How would you find the length of a vector in a parallelepiped?
5. What other properties of vectors can be inferred from the symmetry of a parallelepiped?

Tip: When working with vectors in 3D shapes, symmetry and parallelism often simplify complex relationships.