If the angle between `veca` and `vecb` is `(pi)/(3)`, then angle between `2veca` and `-3vecb` is :

Options

A. `pi//3`
B. `2pi//3`
C. `pi//6`
D. `5pi//3`

Explanation

To find the angle between the vectors \(2\vec{a}\) and \(-3\vec{b}\) given that the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\).

2. **Understand the Transformation of Vectors**:
- The vector \(2\vec{a}\) is simply \(\vec{a}\) scaled by a factor of 2. This means the direction of \(2\vec{a}\) remains the same as that of \(\vec{a}\).
- The vector \(-3\vec{b}\) is the vector \(\vec{b}\) scaled by -3. This means the magnitude is 3 times that of \(\vec{b}\), but the direction is reversed.

3. **Determine the New Angle**:
- The angle between \(2\vec{a}\) and \(-3\vec{b}\) can be found by considering the original angle between \(\vec{a}\) and \(\vec{b}\).
- Since \(-3\vec{b}\) is in the opposite direction of \(\vec{b}\), the angle between \(2\vec{a}\) and \(-3\vec{b}\) will be the supplementary angle to the original angle.
- The supplementary angle can be calculated as:
\[ \text{New Angle} = \pi - \frac{\pi}{3} \]

4. **Calculate the New Angle**:
- Simplifying the expression:
\[ \text{New Angle} = \pi - \frac{\pi}{3} = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

5. **Conclusion**:
- The angle between \(2\vec{a}\) and \(-3\vec{b}\) is \(\frac{2\pi}{3}\).

### Final Answer:
The angle between \(2\vec{a}\) and \(-3\vec{b}\) is \(\frac{2\pi}{3}\).

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