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.