https://file.aiquickdraw.com/mathbot/file/1722739747217mjqvcrwi.jpg

Show a complete solution path being sure to present your work in a complete and organized manner. Show all steps in the solution process. Include a sketch showing F1, F2 and the resultant. Show all steps in the addition and computation of the resultant

The diagram above shows the vector addition of \( F_1 \) and \( F_2 \) to produce the resultant force \( \mathbf{R} \). Here's a summary of the solution:

1. **Forces Identified**:
   - \( F_1 = 2\mathbf{i} + 3\mathbf{j} \)
   - \( F_2 = \mathbf{i} + \mathbf{j} \)

2. **Resultant Force Calculation**:
   - \( \mathbf{R} = F_1 + F_2 = 3\mathbf{i} + 4\mathbf{j} \)

3. **Magnitude of Resultant Force**:
   - \( |\mathbf{R}| = 5 \)

4. **Direction of Resultant Force**:
   - \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \)

5. **Sketch Explanation**:
   - The red vector represents \( F_1 \).
   - The blue vector represents \( F_2 \).
   - The green vector is the resultant \( \mathbf{R} \), showing the combined effect of \( F_1 \) and \( F_2 \).

If you have any further questions or want more details, feel free to ask!

Here are some related questions you might consider:
1. How would you calculate the resultant if the vectors were in three dimensions?
2. What would change if \( F_2 \) were in the opposite direction?
3. How do you resolve vectors into components in non-orthogonal systems?
4. What is the physical significance of the magnitude and direction of the resultant vector?
5. How can you use vector addition to solve problems in real-world applications?
6. What trigonometric identities can simplify the process of finding the direction of a vector?
7. How does the parallelogram law relate to vector addition?
8. How can this method of vector addition be applied to electric fields or forces?

**Tip**: When dealing with vector addition, always ensure the vectors are in the same coordinate system before performing operations.

Explore how to calculate the resultant force using vector addition. Understand the concepts of vector addition, magnitude, and direction of vectors with a detailed example. Suitable for high school students learning physics and mathematics.

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