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

To find the resultant vector of two forces acting at an angle, we can use vector addition. The forces can be broken down into their components and then added to find the resultant force.

### Given:
- Force \( F_1 = 10 \, \text{N} \) acting east.
- Force \( F_2 = 10 \, \text{N} \) acting 70° north of east.

### Steps:

1. **Resolve the Forces into Components:**

   - \( F_1 \) has only an eastward (x-axis) component:
     \[
     F_{1x} = 10 \, \text{N}, \quad F_{1y} = 0 \, \text{N}
     \]

   - \( F_2 \) can be resolved into:
     \[
     F_{2x} = 10 \cos(70^\circ) \, \text{N}, \quad F_{2y} = 10 \sin(70^\circ) \, \text{N}
     \]

     Calculating these:
     \[
     F_{2x} = 10 \times 0.3420 = 3.42 \, \text{N}
     \]
     \[
     F_{2y} = 10 \times 0.9397 = 9.40 \, \text{N}
     \]

2. **Add the Components:**
   \[
   \text{Resultant} \, F_x = F_{1x} + F_{2x} = 10 \, \text{N} + 3.42 \, \text{N} = 13.42 \, \text{N}
   \]
   \[
   \text{Resultant} \, F_y = F_{2y} = 9.40 \, \text{N}
   \]

3. **Find the Magnitude of the Resultant Vector:**
   \[
   R = \sqrt{(F_x)^2 + (F_y)^2}
   \]
   \[
   R = \sqrt{(13.42 \, \text{N})^2 + (9.40 \, \text{N})^2}
   \]
   \[
   R = \sqrt{180.13 + 88.36} \, \text{N}
   \]
   \[
   R = \sqrt{268.49} \, \text{N} \approx 16.39 \, \text{N}
   \]

4. **Find the Direction of the Resultant Vector:**
   \[
   \theta = \tan^{-1}\left(\frac{F_y}{F_x}\right)
   \]
   \[
   \theta = \tan^{-1}\left(\frac{9.40 \, \text{N}}{13.42 \, \text{N}}\right)
   \]
   \[
   \theta \approx \tan^{-1}(0.70) \approx 35^\circ \, \text{north of east}
   \]

### Result:
The resultant vector has a magnitude of approximately **16.39 N** and is directed **35° north of east**.

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Would you like a more detailed explanation, or do you have any other questions?

### Related Questions:
1. What happens to the resultant vector if both forces act in the same direction?
2. How do you resolve a vector into components?
3. Can the method used here apply to more than two vectors?
4. How would the resultant force change if one of the forces increased?
5. What is the physical significance of the resultant vector?
6. How can this method be extended to forces in three dimensions?
7. What is the role of trigonometry in vector addition?
8. How does the angle between vectors affect the magnitude of the resultant vector?

### Tip:
Always make sure to check the angle orientation when resolving vectors into components; it helps to correctly determine the direction of the resultant vector.

Learn how to find the resultant vector of two forces: 10 N each, where one acts east and the other 70 degrees north of east. Detailed step-by-step solution using vector addition and trigonometry. Suitable for high school students.

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