In this problem, you are asked to evaluate several vector equations based on the figure showing a three-dimensional rectangular box. Let's analyze the statements step by step by considering the relationships between the vectors in the diagram.

Here are the vectors shown and their likely orientations based on the axes:

• $\mathbf{c}$ is in the $xz$-plane.
• $\mathbf{e}$ is in the $xy$-plane.
• The vectors $\mathbf{a}, \mathbf{b}, \mathbf{d}, \mathbf{f}, \mathbf{g}$ are oriented within the 3D space based on their labels and positions in the diagram.

Now, let's check each statement:

1. $\mathbf{c} = \mathbf{f}$
   This equation checks if vector $\mathbf{c}$ is equal to vector $\mathbf{f}$. Based on the image, vectors $\mathbf{c}$ and $\mathbf{f}$ appear to be located on different edges of the rectangular box. Therefore, it seems unlikely that $\mathbf{c} = \mathbf{f}$.
   False

2. $\mathbf{g} = \mathbf{f} + \mathbf{a}$
   This is a vector addition problem. Vector $\mathbf{g}$ seems to span from the origin of the box to the top front corner, while vectors $\mathbf{f}$ and $\mathbf{a}$ also connect various vertices of the box. By comparing directions, it's possible that the vector addition $\mathbf{f} + \mathbf{a}$ matches $\mathbf{g}$.
   True

3. $\mathbf{d} = \mathbf{g} - \mathbf{c}$
   Here, the equation involves subtracting vectors. The vector $\mathbf{g}$ reaches the top front of the box, and subtracting vector $\mathbf{c}$, which lies in the $xz$-plane, should lead to vector $\mathbf{d}$. Based on the geometry, this relationship appears plausible.
   True

4. $\mathbf{a} = -\mathbf{b}$
   This equation checks if vector $\mathbf{a}$ is the opposite of vector $\mathbf{b}$. Looking at their directions, vectors $\mathbf{a}$ and $\mathbf{b}$ do not seem to be opposites of each other.
   False

5. $\mathbf{a} = \mathbf{d}$
   This equation suggests that vector $\mathbf{a}$ and vector $\mathbf{d}$ are the same. However, these vectors appear to occupy different edges of the box, making this relationship unlikely.
   False

6. $\mathbf{e} = \mathbf{a} - \mathbf{b}$
   Vector $\mathbf{e}$ lies in the $xy$-plane, and $\mathbf{a} - \mathbf{b}$ seems to create a vector that points in the same direction as $\mathbf{e}$. Based on the diagram, this seems reasonable.
   True

Final answers:

1. False
2. True
3. True
4. False
5. False
6. True

Would you like further details or clarification on any of the steps? Here are five related questions you might find useful:

1. How do you perform vector addition and subtraction geometrically?
2. What properties do equal vectors need to share?
3. How can you determine if two vectors are opposites?
4. What are the conditions for a vector equation to hold true in 3D space?
5. How do projections onto specific planes affect the properties of vectors?

Tip: Always remember that vector equality means that both magnitude and direction must be identical, regardless of their starting points.