What is the angle between 3a and -5a? What is the ratio of magnitude of two vectors?

Answer

To solve the problem, we need to find the angle between the vectors \(3\mathbf{a}\) and \(-5\mathbf{a}\), as well as the ratio of their magnitudes.

### Step-by-Step Solution:

1. **Identify the Vectors**:
- We have two vectors: \( \mathbf{A} = 3\mathbf{a} \) and \( \mathbf{B} = -5\mathbf{a} \).

2. **Determine the Direction**:
- The vector \(3\mathbf{a}\) points in the direction of \(\mathbf{a}\).
- The vector \(-5\mathbf{a}\) points in the opposite direction to \(\mathbf{a}\).

3. **Visualize the Vectors**:
- When we visualize these vectors, \(3\mathbf{a}\) points to the right (for example), and \(-5\mathbf{a}\) points to the left.

4. **Calculate the Angle**:
- The angle between two vectors that point in opposite directions is \(180^\circ\).
- Therefore, the angle between \(3\mathbf{a}\) and \(-5\mathbf{a}\) is \(180^\circ\).

5. **Calculate the Magnitudes**:
- The magnitude of \(3\mathbf{a}\) is \(3\).
- The magnitude of \(-5\mathbf{a}\) is \(5\) (the negative sign does not affect the magnitude).

6. **Find the Ratio of Magnitudes**:
- The ratio of the magnitudes of the two vectors is given by:
\[ \text{Ratio} = \frac{\text{Magnitude of } 3\mathbf{a}}{\text{Magnitude of } -5\mathbf{a}} = \frac{3}{5} \]

### Final Answers:
- The angle between \(3\mathbf{a}\) and \(-5\mathbf{a}\) is \(180^\circ\).
- The ratio of the magnitudes of the two vectors is \(\frac{3}{5}\).