A force vector of 45.0 N @ 30.0° north of east is added to another force vector 75.0 N [N]. What is the resulting force?

To solve the problem of finding the resulting force when two vectors are added, we can break it down into components. Let's start by resolving the force vectors into their respective \(x\) (east) and \(y\) (north) components:

### 1. Resolve the first vector (45.0 N at 30.0° north of east):
- **\(x\)-component (east direction):**
  \[
  F_{1x} = 45.0 \, \text{N} \times \cos(30.0^\circ) = 45.0 \, \text{N} \times \frac{\sqrt{3}}{2} \approx 38.97 \, \text{N}
  \]
  
- **\(y\)-component (north direction):**
  \[
  F_{1y} = 45.0 \, \text{N} \times \sin(30.0^\circ) = 45.0 \, \text{N} \times \frac{1}{2} = 22.5 \, \text{N}
  \]

### 2. Resolve the second vector (75.0 N directly north):
- Since this force is directed entirely north, it has no \(x\)-component and only a \(y\)-component:
  \[
  F_{2x} = 0 \, \text{N}
  \]
  \[
  F_{2y} = 75.0 \, \text{N}
  \]

### 3. Add the components of the two vectors:
- **Resulting \(x\)-component:**
  \[
  F_x = F_{1x} + F_{2x} = 38.97 \, \text{N} + 0 \, \text{N} = 38.97 \, \text{N}
  \]

- **Resulting \(y\)-component:**
  \[
  F_y = F_{1y} + F_{2y} = 22.5 \, \text{N} + 75.0 \, \text{N} = 97.5 \, \text{N}
  \]

### 4. Find the magnitude of the resulting force vector:
\[
F_{\text{resultant}} = \sqrt{F_x^2 + F_y^2} = \sqrt{(38.97 \, \text{N})^2 + (97.5 \, \text{N})^2} \approx \sqrt{1517.76 + 9506.25} \, \text{N} \approx \sqrt{11024.01} \, \text{N} \approx 104.98 \, \text{N}
\]

### 5. Determine the direction of the resulting force vector:
The angle \( \theta \) north of east is given by:
\[
\theta = \tan^{-1} \left(\frac{F_y}{F_x}\right) = \tan^{-1} \left(\frac{97.5 \, \text{N}}{38.97 \, \text{N}}\right) \approx \tan^{-1} (2.5) \approx 68.2^\circ
\]

### Final Answer:
The resulting force vector has a magnitude of approximately **105.0 N** and is directed at **68.2° north of east**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you calculate the resultant of three or more vectors?
2. What happens when you add perpendicular force vectors?
3. How do you resolve a vector into components along different axes?
4. How does vector addition differ from scalar addition?
5. How do forces combine in non-right-angled vector addition?

**Tip:** Always double-check the units and angles when resolving vectors to ensure accuracy.

Calculate the resultant force when a 45.0 N vector at 30.0° north of east is added to a 75.0 N northward vector. This solution involves vector addition, trigonometry, and the Pythagorean theorem, suitable for students in Grades 10-12.

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