It looks like you've uploaded a question related to a 3D rectangular box with several vectors. Let me summarize the given image and analyze the statements provided:

• The box is in a 3D space with several vectors labeled in blue: $\vec{a}, \vec{b}, \vec{c}, \vec{d}, \vec{e}, \vec{f}, \vec{g}$.
• $\vec{c}$ lies in the $xz$-plane, and $\vec{e}$ lies in the $xy$-plane.

Let's now evaluate the truth or falsity of the statements based on the image.

1. $\vec{a} = \vec{d}$:
   This is false because vector $\vec{a}$ points in a different direction than vector $\vec{d}$. The positions and orientations of these vectors are not identical.

2. $\vec{c} = \vec{f}$:
   This is true. The vectors $\vec{c}$ and $\vec{f}$ seem to point in the same direction based on the image.

3. $\vec{a} = -\vec{b}$:
   This is false. $\vec{a}$ and $\vec{b}$ do not appear to be opposite in direction; they have different orientations.

4. $\vec{e} = \vec{a} - \vec{b}$:
   This is false. Based on the geometric positioning, $\vec{e}$ does not directly result from subtracting $\vec{a}$ and $\vec{b}$.

5. $\vec{d} = \vec{g} - \vec{c}$:
   This is true. The vector $\vec{d}$ is in the correct position relative to $\vec{g}$ and $\vec{c}$, matching this statement.

6. $\vec{g} = \vec{f} + \vec{a}$:
   This is true. Based on the vectors' positions, $\vec{g}$ indeed appears to be the sum of $\vec{f}$ and $\vec{a}$.

Let me know if you'd like any further clarification or if you have more questions about vector analysis!

Here are some related questions to explore further:

1. How do you determine the vector sum in a 3D space?
2. How can we check if two vectors are equal in direction and magnitude?
3. What geometric properties do vectors have in 3D space?
4. Can vector subtraction be represented geometrically, and if so, how?
5. What is the role of unit vectors in defining directions in 3D space?

Tip: In vector algebra, breaking down vectors into components along the coordinate axes can help in determining their relationships and simplifying calculations.