If the angle between the vectors `vec(a) and vec(b)` is `(pi)/(3)`. Then what is the angle between `-5 vec(a) and 6 vec(b)` ?

Options

A. `(pi)/(6)`
B. `(2pi)/(3)`
C. `(2pi)/(5)`
D. `(3pi)/(7)`

Answer

To find the angle between the vectors \(-5\vec{a}\) and \(6\vec{b}\) given that the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\), we can follow these steps:

### Step 1: Understand the Given Information
We know that the angle between the vectors \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\) radians, which is equivalent to \(60^\circ\).

### Step 2: Analyze the Effect of Scalar Multiplication
When we multiply a vector by a scalar, the direction of the vector may change (if the scalar is negative), but the angle between the vectors remains unchanged. Therefore, the angle between \(\vec{a}\) and \(6\vec{b}\) is still \(\frac{\pi}{3}\) (or \(60^\circ\)).

### Step 3: Determine the Angle Between \(-5\vec{a}\) and \(6\vec{b}\)
The vector \(-5\vec{a}\) is in the opposite direction of \(5\vec{a}\). When we consider the angle between \(-5\vec{a}\) and \(6\vec{b}\), we can visualize it as follows:
- The angle between \(\vec{a}\) and \(\vec{b}\) is \(60^\circ\).
- The angle between \(-5\vec{a}\) and \(\vec{b}\) is \(180^\circ - 60^\circ = 120^\circ\).

### Step 4: Convert the Angle to Radians
Now, we need to express \(120^\circ\) in radians:
\[ 120^\circ = \frac{120 \times \pi}{180} = \frac{2\pi}{3} \]

### Conclusion
Thus, the angle between the vectors \(-5\vec{a}\) and \(6\vec{b}\) is \(\frac{2\pi}{3}\) radians.