Assertion If angle between a and b is `30^(@)` . Then angle between 2a and `-(b) /(2)` will be `150^(@)`
Reason Sign of dot product of two vectors tells you whether angle between two vectors is acute or obtuse .

Options

A. If both Assetion and Reason are correct but Reason is the correct explanation of Assertion.
B. If both Assetion and Reason are correct but Reason is not the correct explanation of Assertion.
C. If Asserion is true but Reason is false
D. If Asserion is false but Reason is true

Explanation

To solve the problem, we need to analyze the assertion and the reason given in the question.

### Step 1: Understanding the Assertion
The assertion states that if the angle between vectors **a** and **b** is \(30^\circ\), then the angle between \(2\mathbf{a}\) and \(-\frac{\mathbf{b}}{2}\) will be \(150^\circ\).

### Step 2: Finding the Angle Between \(2\mathbf{a}\) and \(-\frac{\mathbf{b}}{2}\)
1. The angle between two vectors can be found using the formula for the dot product:
\[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta) \]
where \(\theta\) is the angle between vectors \(\mathbf{u}\) and \(\mathbf{v}\).

2. In our case, let \(\mathbf{u} = 2\mathbf{a}\) and \(\mathbf{v} = -\frac{\mathbf{b}}{2}\).

3. The angle between \(\mathbf{u}\) and \(\mathbf{v}\) can be calculated as follows:
- The angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(30^\circ\).
- The angle between \(\mathbf{a}\) and \(-\mathbf{b}\) will be \(180^\circ - 30^\circ = 150^\circ\).
- Since scaling a vector by a positive scalar (like 2) does not change the angle, the angle between \(2\mathbf{a}\) and \(-\frac{\mathbf{b}}{2}\) remains \(150^\circ\).

### Conclusion for the Assertion
Thus, the assertion is correct.

### Step 3: Understanding the Reason
The reason states that the sign of the dot product of two vectors tells you whether the angle between them is acute or obtuse.

1. If the dot product \(\mathbf{u} \cdot \mathbf{v} > 0\), the angle \(\theta\) is acute (\(0^\circ < \theta < 90^\circ\)).
2. If the dot product \(\mathbf{u} \cdot \mathbf{v} < 0\), the angle \(\theta\) is obtuse (\(90^\circ < \theta < 180^\circ\)).
3. If the dot product \(\mathbf{u} \cdot \mathbf{v} = 0\), the angle \(\theta\) is \(90^\circ\).

### Conclusion for the Reason
The reason is also correct, as it accurately describes the relationship between the dot product and the angles.

### Final Conclusion
Both the assertion and the reason are correct, but the reason does not explain the assertion directly. Therefore, the correct option is that both are correct, but the reason is not the correct explanation for the assertion.