https://file.aiquickdraw.com/mathbot/file/1729757489913hn8zpboq.jpg

Let's break down and solve each part of the problem from the image.

### Vectors in the Same Direction:
1. **\(-15\text{ m} \rightarrow + 30\text{ m} \rightarrow = ?\)**  
   Since both vectors are in the same direction, you can add them directly:  
   \(-15\text{ m} + 30\text{ m} = 15\text{ m} \rightarrow\)

2. **\(12\text{ cm} \rightarrow + 8\text{ cm} \leftarrow = ?\)**  
   The two vectors are in opposite directions, so we subtract the magnitudes. Convert to the same unit first:
   \[
   12\text{ cm} = 120\text{ mm}
   \quad \text{and} \quad 8\text{ cm} = 80\text{ mm}
   \]
   Since they are in opposite directions:
   \[
   120\text{ mm} - 80\text{ mm} = 40\text{ mm} \rightarrow \text{ (final direction)}
   \]

### Vectors in Opposite Directions:
3. **\(16\text{ mm} \leftarrow + (-17\text{ mm} \rightarrow) = ?\)**  
   These vectors are in opposite directions. We can simply subtract their magnitudes:
   \[
   16\text{ mm} - 17\text{ mm} = -1\text{ mm}
   \]
   The negative sign means the resultant vector points to the right, so the result is:
   \[
   1\text{ mm} \rightarrow
   \]

4. **\(32\text{ m} \downarrow + (-16\text{ m} \uparrow) = ?\)**  
   The vectors are in opposite vertical directions:
   \[
   32\text{ m} - 16\text{ m} = 16\text{ m} \downarrow
   \]

### Vectors in Perpendicular Directions:
5. **\(X^2 + Y^2 = R^2\)**  
   For this, we use the Pythagorean theorem to find the resultant of the perpendicular vectors:
   \[
   \text{Vector 1: } 12\text{ dam} \rightarrow
   \quad \text{and} \quad \text{Vector 2: } 8\text{ dam} \uparrow
   \]
   Using the Pythagorean theorem:
   \[
   R^2 = (12^2 + 8^2) = 144 + 64 = 208
   \]
   So, \(R = \sqrt{208} \approx 14.42\text{ dam}\).

   You would then graph this with a horizontal vector of \(12\text{ dam} \rightarrow\) and a vertical vector of \(8\text{ dam} \uparrow\).

Would you like more details or further explanations on any of these problems?

Here are some related questions to explore:
1. What is the significance of direction in vector addition?
2. How can you determine the resultant vector when dealing with multiple directions?
3. Why do you use the Pythagorean theorem for perpendicular vectors?
4. How would vector subtraction differ if the vectors were in the same direction?
5. How does changing the unit (e.g., from cm to mm) impact vector calculations?

**Tip:** Always ensure all vectors are in the same units before performing operations like addition or subtraction.

Solve vector problems involving addition and subtraction in the same, opposite, and perpendicular directions.

Learn how to solve vector problems involving addition and subtraction of vectors in the same, opposite, and perpendicular directions. Includes the use of the Pythagorean theorem to find resultant vectors. Suitable for high school physics students.

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