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To solve problem 12, we need to find the required angle \(\theta\) such that the resultant force \(R\) of two forces (600 N and 900 N) is vertical. We also need to find the corresponding magnitude of \(R\).

### Given:
- Force \(P = 600 \text{ N}\)
- Second force = \(900 \text{ N}\)
- Angle between \(P\) and the horizontal is \(45^\circ\)

### Step-by-Step Solution:

1. **Resolve forces into components:**
   To make the resultant force vertical, the horizontal components of the forces should cancel out.

   - Resolve the 600 N force (\(P\)) into horizontal and vertical components:
     \[
     P_x = 600 \cos(45^\circ)
     \]
     \[
     P_y = 600 \sin(45^\circ)
     \]

   - Resolve the 900 N force into components using the unknown angle \(\theta\):
     \[
     900 \cos(\theta)
     \]
     \[
     900 \sin(\theta)
     \]

2. **Set the horizontal components to cancel out:**
   Since the resultant is vertical, the sum of the horizontal components must be zero:
   \[
   600 \cos(45^\circ) - 900 \cos(\theta) = 0
   \]

   Solving for \(\theta\):
   \[
   \cos(\theta) = \frac{600 \cos(45^\circ)}{900}
   \]
   \[
   \cos(\theta) = \frac{600 \times \frac{\sqrt{2}}{2}}{900}
   \]
   \[
   \cos(\theta) = \frac{600 \times 0.707}{900} \approx 0.471
   \]
   \[
   \theta = \cos^{-1}(0.471) \approx 61.3^\circ
   \]

3. **Find the magnitude of the vertical resultant:**
   The vertical components add up to give the resultant force:
   \[
   R = 600 \sin(45^\circ) + 900 \sin(61.3^\circ)
   \]
   \[
   R = 600 \times 0.707 + 900 \times 0.874
   \]
   \[
   R \approx 424.2 + 786.6 = 1210.8 \text{ N}
   \]

Thus, the required angle \(\theta\) is approximately \(61.3^\circ\), and the corresponding magnitude of the resultant force \(R\) is about \(1210.8 \text{ N}\).

Would you like further details on this solution, or do you have other questions?

### Related Questions:
1. How do we resolve a force into horizontal and vertical components?
2. What is the significance of setting the horizontal components to zero for a vertical resultant?
3. Can we use a different method to find the resultant force angle?
4. What if the forces were not inclined at 45 degrees and \(\theta\)? How would that change the calculations?
5. How can we find the resultant direction if the forces do not add up to a vertical resultant?

### Tip:
When solving vector problems, sketching the forces and components can greatly aid in visualizing the solution and ensuring that calculations are correctly set up.

Determine the required angle θ if the resultant R of the two forces (600 N and 900 N) is to be vertical. Find the corresponding value of R.

Solve for the required angle θ and the resultant force R when two forces, 600 N and 900 N, combine to create a vertical resultant. Includes detailed steps on vector resolution and balancing horizontal components.

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