To solve for the maximum and minimum possible values of the magnitude of the resultant vector $\mathbf{R}$, we can use the following vector addition principles:

1. Maximum Magnitude of $R$: The maximum magnitude occurs when vectors $\mathbf{A}$ and $\mathbf{B}$ are in the same direction (i.e., the angle between them is $0^\circ$). In this case, the resultant vector's magnitude is simply the sum of the magnitudes of $\mathbf{A}$ and $\mathbf{B}$:

   $R_{\text{max}} = |\mathbf{A}| + |\mathbf{B}| = 3\, \text{m} + 8\, \text{m} = 11\, \text{m}$

2. Minimum Magnitude of $R$: The minimum magnitude occurs when vectors $\mathbf{A}$ and $\mathbf{B}$ are in opposite directions (i.e., the angle between them is $180^\circ$). In this case, the resultant vector's magnitude is the difference between the magnitudes of $\mathbf{A}$ and $\mathbf{B}$:

   $R_{\text{min}} = |\mathbf{B}| - |\mathbf{A}| = 8\, \text{m} - 3\, \text{m} = 5\, \text{m}$

Summary:

• (a) Maximum magnitude of $R$: $11\, \text{m}$
• (b) Minimum magnitude of $R$: $5\, \text{m}$

Would you like more details or have any questions?

Here are 5 related questions:

1. What happens if the angle between the vectors is neither $0^\circ$ nor $180^\circ$?
2. How would you calculate the resultant magnitude if the angle between vectors is $90^\circ$?
3. How do you find the direction of the resultant vector?
4. Can two vectors with different magnitudes ever result in a zero resultant vector?
5. What is the geometric representation of vector addition?

Tip: When working with vectors, always consider both magnitude and direction. The angle between vectors significantly affects the resultant.