
Two vectors A and B are added together to give a resultant vector Ř: R = A + B. The magnitudes of A and B are 3 m and 8 m, respectively, but the vectors can have any
orientation. What is (a) the maximum possible value and (b) the minimum possible value for the magnitude of R?

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.

Two vectors A and B are added together to give a resultant vector Ř: R = A + B. The magnitudes of A and B are 3 m and 8 m, respectively, but the vectors can have any orientation. What is (a) the maximum possible value and (b) the minimum possible value for the magnitude of R?

Learn how to find the maximum and minimum magnitude of the resultant vector when adding two vectors with magnitudes of 3 m and 8 m. This problem involves vector addition principles and the triangle inequality theorem. Suitable for students in Grades 9-12.

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