https://file.aiquickdraw.com/mathbot/file/1727967020468136zrib5.jpg

The image provides a geometric representation with three non-coplanar vectors: \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \). We are tasked with expressing the other vectors in the diagram in terms of these three vectors.

### Step-by-step solution:

1. **Understanding the Diagram**:
   - The three given vectors are non-coplanar, meaning they span a three-dimensional space.
   - The vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) appear to define the edges of a parallelepiped.
   - The points marked include \( O \) (the origin), and points along various lines, such as \( P \), \( Q \), \( R \), etc.

2. **Key relationships in the figure**:
   - \( \mathbf{OA} = \mathbf{a} \)
   - \( \mathbf{OB} = \mathbf{b} \)
   - \( \mathbf{OC} = \mathbf{c} \)

3. **Expressing other vectors**:
   - **Vector \( \mathbf{OP} \)**: Since \( P \) lies on the same line as \( B \), we express \( \mathbf{OP} \) as some scalar multiple of \( \mathbf{b} \). Therefore, \( \mathbf{OP} = k\mathbf{b} \), where \( k \) is some scalar.
   
   - **Vector \( \mathbf{OQ} \)**: \( Q \) seems to lie along the plane spanned by \( \mathbf{a} \) and \( \mathbf{b} \). Thus, \( \mathbf{OQ} \) is a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \), i.e., \( \mathbf{OQ} = m\mathbf{a} + n\mathbf{b} \).
   
   - **Vector \( \mathbf{OR} \)**: Since \( R \) appears to be a point in the interior of the parallelepiped, it can be expressed as a combination of \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \). So, \( \mathbf{OR} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c} \).
   
   - **Vector \( \mathbf{OU} \)**: Similarly, \( U \) seems to be a point involving a combination of all three vectors. We can express \( \mathbf{OU} = s\mathbf{a} + t\mathbf{b} + u\mathbf{c} \).

### Conclusion:
Each vector can be expressed as:
- \( \mathbf{OP} = k\mathbf{b} \)
- \( \mathbf{OQ} = m\mathbf{a} + n\mathbf{b} \)
- \( \mathbf{OR} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c} \)
- \( \mathbf{OU} = s\mathbf{a} + t\mathbf{b} + u\mathbf{c} \)

Please confirm the relationships or let me know if additional clarification is needed on any of the vectors.

Would you like further details or clarifications on any step?

### Related questions:
1. How can we use the cross product to verify non-coplanarity of vectors?
2. What conditions would make three vectors coplanar?
3. How can you find the volume of the parallelepiped formed by vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \)?
4. How do scalar multiples of vectors change their magnitude and direction?
5. How would you solve for the scalars \( p, q, r \) given specific points on the parallelepiped?

### Tip:
Understanding how to express vectors in terms of a basis helps in analyzing vector spaces and geometric objects in higher dimensions!

Diberikan tiga vektor non-koplanar a, b, dan c. Carilah ekspresi untuk setiap vektor dalam ruang tiga dimensi berdasarkan gambar yang diberikan.

Learn how to express vectors in 3D space using non-coplanar vectors a, b, and c with a detailed diagram. This problem explores key concepts in vector algebra such as linear combinations and the representation of points in three-dimensional space.

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