If `theta` be the angle between two vectors `vec(a)` and `vec(b)`, then `vec(a).vec(b)ge0` only when

Options

A. `0ltthetaltpi/2`
B. `0lethetalepi/2`
C. `0ltthetaltpi`
D. `0lethetalepi`

Answer

To solve the problem, we need to determine the conditions under which the dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is greater than or equal to zero.

### Step-by-Step Solution:

1. **Understand the Dot Product**: The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is given by the formula:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
where \(\theta\) is the angle between the two vectors.

2. **Set Up the Inequality**: We are given the condition:
\[ \vec{a} \cdot \vec{b} \geq 0 \]
Substituting the dot product formula, we have:
\[ |\vec{a}| |\vec{b}| \cos \theta \geq 0 \]

3. **Analyze the Components**: Since the magnitudes \(|\vec{a}|\) and \(|\vec{b}|\) are always non-negative (they are lengths), the sign of the dot product depends entirely on \(\cos \theta\):
\[ \cos \theta \geq 0 \]

4. **Determine the Range of \(\theta\)**: The cosine function is non-negative in the following intervals:
- From \(0\) to \(\frac{\pi}{2}\) (first quadrant)
- From \(\frac{3\pi}{2}\) to \(2\pi\) (fourth quadrant)

However, since we are interested in the angle between two vectors, we typically restrict \(\theta\) to the range:
\[ 0 \leq \theta \leq \frac{\pi}{2} \]

5. **Conclusion**: Therefore, the condition for \(\vec{a} \cdot \vec{b} \geq 0\) is satisfied when:
\[ 0 \leq \theta \leq \frac{\pi}{2} \]

### Final Answer:
The angle \(\theta\) between the vectors \(\vec{a}\) and \(\vec{b}\) must satisfy:
\[ 0 \leq \theta \leq \frac{\pi}{2} \]