The question is asking to identify the correct relationship between the vectors $\vec{a}$, $\vec{u}$, and $\vec{n}$ shown in the diagram. Here’s a breakdown of the possible relationships based on the given options.

Analysis

1. Vector Relationships:

   • The vector $\vec{n}$ seems to be perpendicular to a line (possibly the $\varepsilon$ line).
   • $\vec{u}$ has a component along $\vec{n}$ as well as a component that is not along $\vec{n}$.
   • $\vec{a}$ appears to be the projection or some combination of $\vec{u}$ and $\vec{n}$.

2. Answer Options:

   • Each answer option uses either a projection formula or cross product, typically to resolve $\vec{a}$ in terms of $\vec{u}$ and $\vec{n}$:
     • $\vec{a} = \frac{\vec{u} \times \vec{n}}{\|\vec{n}\|^2} \times \vec{n}$
     • $\vec{a} = \frac{\vec{u} \cdot \vec{n}}{\|\vec{n}\|^2} \vec{n}$
     • $\vec{a} = \vec{u} - \frac{\vec{u} \cdot \vec{n}}{\|\vec{n}\|^2} \vec{n}$
     • $\vec{a} = \frac{\vec{u} \cdot \vec{n}}{\|\vec{n}\|} \vec{n}$

3. Key Observations:

   • If $\vec{a}$ is the projection of $\vec{u}$ onto $\vec{n}$, the formula would be: $\vec{a} = \frac{\vec{u} \cdot \vec{n}}{\|\vec{n}\|^2} \vec{n}$
   • If $\vec{a}$ represents the component of $\vec{u}$ perpendicular to $\vec{n}$, it would be: $\vec{a} = \vec{u} - \frac{\vec{u} \cdot \vec{n}}{\|\vec{n}\|^2} \vec{n}$

From the visual cue and the list of options, the correct relationship likely corresponds to the projection formula, especially if $\vec{a}$ aligns with $\vec{n}$ direction.

Conclusion

The correct answer is likely: $\vec{a} = \frac{\vec{u} \cdot \vec{n}}{\|\vec{n}\|^2} \vec{n}$

Would you like a deeper explanation of the projection concept or further clarification on vector operations?

Follow-up Questions

1. What is a projection in vector mathematics?
2. How do cross products differ from dot products in calculating vector relationships?
3. How is a vector component along a direction determined?
4. What does it mean geometrically for two vectors to be perpendicular?
5. How are vector magnitudes calculated in Euclidean space?

Tip

When finding a vector's component in a certain direction, use the dot product to calculate the projection efficiently.