https://file.aiquickdraw.com/mathbot/file/1726701599497fzqaazc6.jpg

The vector R~ shown in the sketch may be expressed in terms of A~ , B~ , C~ , and D~ , which are the edges of a parallelogram, as 1. R~ = A~ + D . ~ 2. R~ = D~ − A . ~ 3. R~ = A~ − D .4. R~ = A~ − C . ~ 5. R~ = A~ − B . ~ 6. R~ = C~ + B . ~ 7. R~ = B~ + A . ~ 8. R~ = B~ + D . ~ 9. R~ = C~ + D . ~ 10. R~ = B~ − A . ~

To analyze this vector diagram and express the vector **R** in terms of the vectors **A**, **B**, **C**, and **D**, I'll examine the relationships between the vectors in the parallelogram and how **R** fits into this structure.

### Step-by-step breakdown:
1. The diagram shows a parallelogram, where vectors **A**, **B**, **C**, and **D** represent its edges.
2. The vector **R** is inside the parallelogram and seems to connect two points within the figure, indicating it could be a combination of the vectors **A**, **B**, **C**, and **D**.
3. The vectors **P** and **R** seem to follow geometric vector rules, meaning we can apply vector addition or subtraction.

By examining the relative directions, here's how the relationships look:

- **A** and **C** represent the sides of the parallelogram in one direction.
- **B** and **D** represent the sides in the opposite direction.
- **R** appears to span diagonally across the parallelogram.

Now, let’s check which equation describes **R** correctly based on this geometry:
- **Equation 6: R = C + B** looks plausible since **R** seems to follow the path of combining **C** and **B** in the direction they point. This is due to the structure of the parallelogram.

### Conclusion:
The correct expression for **R** based on the diagram is:
\[ \mathbf{R} = \mathbf{C} + \mathbf{B} \]

Would you like a more detailed explanation of how the vector relationships were analyzed?

### 5 Related Questions:
1. How does vector addition work geometrically in a parallelogram?
2. What is the parallelogram law of vector addition?
3. How can you express a diagonal vector in terms of its sides in any quadrilateral?
4. How would the relationship change if **R** were outside the parallelogram?
5. What other vector operations can be applied in vector analysis of polygons?

### Tip:
When dealing with vector diagrams in geometry, always verify the direction of vectors to ensure accurate addition or subtraction.

This problem shows how to express vector R in terms of vectors A, B, C, and D using the parallelogram law of vector addition. Learn the correct vector equation and understand how R is formed through geometric relationships.

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