https://file.aiquickdraw.com/mathbot/file/17262017529135oqxj4gi.jpg

In this type of question, you are being asked to label the axes of a coordinate system to ensure it follows the right-handed rule. The right-handed coordinate system is a standard way of orienting axes in 3D space. To apply the right-handed rule, point your right-hand fingers along the positive \(x\)-axis, curl them towards the positive \(y\)-axis, and your thumb will point in the direction of the positive \(z\)-axis.

### Steps:
1. **Determine the directions of the given axes**: 
   - One axis is downward, which could represent the \(z\)-axis.
   - The two horizontal axes are likely to be \(x\) and \(y\).

2. **Apply the right-hand rule**: 
   - Align your right-hand fingers with the \(x\)-axis, curl towards \(y\), and the thumb should indicate the \(z\)-axis direction.
   
Looking at the visual cues in the image (arrows and their directions) and the right-handed rule, you should arrange the axes labels to follow this convention:

- Assign the **downward** axis as \(z\) (as commonly done in physics when dealing with gravity or vertical systems).
- Then label the horizontal axes to align with the right-handed system.

If you need any more specific help on the diagram, feel free to ask!

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Here are some follow-up questions you might find helpful:
1. How does the right-handed rule relate to coordinate system orientations in 3D space?
2. Why is it important to use the right-handed system in physics and engineering?
3. Can you identify situations where the left-handed coordinate system is used instead?
4. How would you apply the right-handed rule in a rotated or skewed coordinate system?
5. How can the choice of axis orientation simplify problem-solving in fluid mechanics?

**Tip:** When dealing with coordinate systems, always double-check your orientation using the right-handed rule to avoid confusion in 3D vector operations.

Label the axes of a coordinate system so that each follows a right-handed rule, given one axis is directed downward.

Learn how to label the axes in a 3D coordinate system to follow the right-handed rule, often used in physics and engineering. This problem helps visualize spatial relationships and how to apply the right-hand rule to align the x, y, and z axes.

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