Two vectors A  and B  are added together to give a resultant vector  R.   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 for the magnitude of  R.
(b) the minimum possible value for the magnitude of  R.  ?

Given two vectors \( \mathbf{A} \) and \( \mathbf{B} \) with magnitudes \( |\mathbf{A}| = 3 \, \text{m} \) and \( |\mathbf{B}| = 8 \, \text{m} \), and knowing that the resultant vector \( \mathbf{R} \) is formed by adding these two vectors, we need to find:

### (a) The maximum possible value for the magnitude of \( \mathbf{R} \)

The maximum magnitude of \( \mathbf{R} \) occurs when the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are aligned in the same direction. In this case, the magnitudes simply add up:

\[
|\mathbf{R}_{\text{max}}| = |\mathbf{A}| + |\mathbf{B}| = 3 \, \text{m} + 8 \, \text{m} = 11 \, \text{m}
\]

### (b) The minimum possible value for the magnitude of \( \mathbf{R} \)

The minimum magnitude of \( \mathbf{R} \) occurs when the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are aligned in opposite directions. In this case, the magnitude of \( \mathbf{R} \) is the difference between the magnitudes of \( \mathbf{A} \) and \( \mathbf{B} \):

\[
|\mathbf{R}_{\text{min}}| = ||\mathbf{B}| - |\mathbf{A}| | = |8 \, \text{m} - 3 \, \text{m}| = 5 \, \text{m}
\]

### Final Answers:
- (a) The maximum possible value for the magnitude of \( \mathbf{R} \) is \( 11 \, \text{m} \).
- (b) The minimum possible value for the magnitude of \( \mathbf{R} \) is \( 5 \, \text{m} \).

Would you like further details on how vector addition works? Or have any more questions?

Here are five related questions:
1. What happens to the resultant vector if the angle between \( \mathbf{A} \) and \( \mathbf{B} \) is 90 degrees?
2. How do you calculate the magnitude of the resultant vector using the law of cosines?
3. What is the direction of the resultant vector when \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular?
4. How do you find the components of the resultant vector in two dimensions?
5. What is the effect of vector subtraction on the resultant vector's magnitude?

**Tip:** When adding vectors, the angle between them significantly affects the magnitude of the resultant vector.

Explore how to find the maximum and minimum magnitude of the resultant vector formed by adding two vectors with given magnitudes (3 m and 8 m). Understand vector addition principles, the law of cosines, and how vector orientations affect resultant vector magnitudes. Suitable for high school students learning vectors and vector addition.

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