Two forces of 10 N each are acting on a point. One force acts east and the other 70° north of east. Find the resultant vector.

To find the resultant vector of the two forces, we'll use vector addition. Here's the step-by-step process:

### Step 1: Break the forces into components

- **Force 1 (East):**  
  - Magnitude: \( F_1 = 10 \, \text{N} \)  
  - Direction: East (0°)  
  - Components:  
    - \( F_{1x} = 10 \, \text{N} \) (since it's entirely in the east direction)  
    - \( F_{1y} = 0 \, \text{N} \)

- **Force 2 (70° North of East):**  
  - Magnitude: \( F_2 = 10 \, \text{N} \)  
  - Direction: 70° north of east  
  - Components:  
    - \( F_{2x} = 10 \cos(70°) \)  
    - \( F_{2y} = 10 \sin(70°) \)

### Step 2: Calculate the components

- \( F_{2x} = 10 \cos(70°) \approx 10 \times 0.3420 = 3.42 \, \text{N} \)
- \( F_{2y} = 10 \sin(70°) \approx 10 \times 0.9397 = 9.40 \, \text{N} \)

### Step 3: Add the components

- **Resultant force in the x-direction:**  
  \( R_x = F_{1x} + F_{2x} = 10 + 3.42 = 13.42 \, \text{N} \)

- **Resultant force in the y-direction:**  
  \( R_y = F_{1y} + F_{2y} = 0 + 9.40 = 9.40 \, \text{N} \)

### Step 4: Calculate the magnitude of the resultant vector

The magnitude \( R \) of the resultant vector is found using the Pythagorean theorem:

\[
R = \sqrt{R_x^2 + R_y^2} = \sqrt{(13.42)^2 + (9.40)^2} \approx \sqrt{180.05 + 88.36} = \sqrt{268.41} \approx 16.39 \, \text{N}
\]

### Step 5: Find the direction of the resultant vector

The direction \( \theta \) of the resultant vector relative to the east direction is given by:

\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{9.40}{13.42}\right) \approx \tan^{-1}(0.70) \approx 35.1^\circ \text{ North of East}
\]

### Final Answer:
The resultant vector has a magnitude of approximately **16.39 N** and is directed **35.1° north of east**.

Would you like more details, or do you have any other questions?

### Related Questions:
1. How do you calculate the resultant force if both forces act in the same direction?
2. What happens if the forces were at an angle of 90° instead of 70°?
3. How do you find the angle between two forces if their magnitudes and resultant are known?
4. What is the significance of resolving forces into components?
5. Can you explain how the sine and cosine functions relate to force components?
6. How would the resultant force change if one force's magnitude was doubled?
7. What is the process to add more than two vectors?
8. How would you calculate the resultant force if the angle was given as 70° south of east?

**Tip:** Understanding vector addition and the resolution of forces into components is crucial for solving problems in physics and engineering involving multiple forces acting on a point.

Learn how to calculate the resultant vector of two forces, 10 Newtons each, where one acts east and the other at 70 degrees north of east. Understand vector addition using components and trigonometric functions. Suitable for high school students and those interested in physics and engineering.

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