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.