If angle between `veca` and `vecb` is `pi/3`, then angle between `veca` and `-3vecb` is

Options

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

Answer

To solve the problem, we need to find the angle between the vector \(\vec{a}\) and the vector \(-3\vec{b}\), given that the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that the angle between vectors \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\).
- We need to find the angle between \(\vec{a}\) and \(-3\vec{b}\).

2. **Visualize the Vectors**:
- Let's visualize \(\vec{a}\) and \(\vec{b}\). The angle between them is \(\frac{\pi}{3}\).
- The vector \(-3\vec{b}\) is simply the vector \(\vec{b}\) scaled by 3 and pointed in the opposite direction.

3. **Determine the Relationship Between Angles**:
- The angle between \(\vec{a}\) and \(-3\vec{b}\) can be determined by considering the angle \(\vec{a}\) makes with \(\vec{b}\) and the fact that \(-3\vec{b}\) is in the opposite direction to \(\vec{b}\).
- If we denote the angle between \(\vec{a}\) and \(-3\vec{b}\) as \(\theta\), we can use the relationship:
\[ \text{Angle between } \vec{a} \text{ and } -3\vec{b} = 180^\circ - \text{Angle between } \vec{a} \text{ and } \vec{b} \]

4. **Calculate the Angle**:
- Since the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\), we can substitute this into our equation:
\[ \theta = \pi - \frac{\pi}{3} \]
- Now, simplify the expression:
\[ \theta = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

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

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

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