https://file.aiquickdraw.com/mathbot/file/1727429630278uue6ciid.jpg

The image asks to sketch the indicated vectors by copying vectors **u**, **v**, and **w** head to tail. These instructions relate to vector addition and subtraction, where vectors are placed head to tail to determine the resultant vectors. Below are the breakdowns for each request:

### a. **u + v**
- **Vector Addition**: Place vector **u** head-to-tail with vector **v**.
- The resulting vector is the diagonal from the tail of **u** to the head of **v**.

### b. **u + v + w**
- Add vectors **u** and **v** as in part (a).
- Then, place vector **w** at the head of the resulting vector **(u + v)**.
- The final result is from the tail of **u** to the head of **w**.

### c. **u - v**
- **Vector Subtraction**: Subtracting **v** is the same as adding **(-v)**.
- Reverse the direction of **v** and then add it to **u** head to tail.
- The resulting vector is the diagonal from the tail of **u** to the head of **(-v)**.

### d. **u - w**
- Similar to part (c), reverse the direction of **w**.
- Place the tail of **u** and the head of **-w** head to tail to find the resulting vector.

If you'd like, I can sketch the vector diagrams for you or provide further clarification.

Would you like more details or have any questions?

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Here are five related questions to expand on this topic:
1. How do you determine the magnitude of a vector when adding two vectors geometrically?
2. What is the difference between vector addition and subtraction?
3. How can you calculate the resultant vector algebraically instead of graphically?
4. How does vector subtraction affect the direction of the resultant vector?
5. What are common applications of vector addition in physics and engineering?

**Tip**: When working with vector subtraction, always remember that subtracting a vector is equivalent to adding its opposite.

Copy vectors u, v, and w head to tail as needed to sketch the indicated vectors: u + v, u + v + w, u - v, u - w.

Learn how to perform vector addition and subtraction geometrically using the head-to-tail method. This problem demonstrates key vector operations like u + v, u + v + w, and u - v, with diagrams for better understanding. Suitable for high school students (Grades 10-12).

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