If `theta` is the angle between two vectors `vec(a)` and `vec(b)`, then `vec(a).vec(b) ge 0` only when

Options

A. `0 lt theta lt ( pi )/(2)`
B. `0 le theta le ( pi)/(2)`
C. `0 lt theta lt pi`
D. `0 le theta le pi `

Answer

To solve the problem, we need to analyze the condition given by the dot product of two vectors \(\vec{a}\) and \(\vec{b}\). The dot product is defined as:

\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]

where \(\theta\) is the angle between the two vectors.

### Step-by-Step Solution:

1. **Understanding the Dot Product Condition**:
We are given that \(\vec{a} \cdot \vec{b} \geq 0\). This implies:
\[ |\vec{a}| |\vec{b}| \cos(\theta) \geq 0 \]

2. **Considering Magnitudes**:
Since \(|\vec{a}|\) and \(|\vec{b}|\) are magnitudes of the vectors, they are always non-negative (i.e., \(|\vec{a}| \geq 0\) and \(|\vec{b}| \geq 0\)). Therefore, the sign of the dot product depends solely on \(\cos(\theta)\).

3. **Analyzing \(\cos(\theta)\)**:
For the inequality \(|\vec{a}| |\vec{b}| \cos(\theta) \geq 0\) to hold, we need:
\[ \cos(\theta) \geq 0 \]

4. **Finding the Range of \(\theta\)**:
The cosine function is non-negative in the following ranges:
- From \(0\) to \(\frac{\pi}{2}\) (first quadrant)
- At \(\theta = 0\) (when the vectors are in the same direction)
- At \(\theta = \frac{\pi}{2}\) (when the vectors are perpendicular)

Therefore, we can conclude:
\[ 0 \leq \theta \leq \frac{\pi}{2} \]

5. **Conclusion**:
The angle \(\theta\) between the two vectors \(\vec{a}\) and \(\vec{b}\) must satisfy:
\[ \theta \in [0, \frac{\pi}{2}] \]

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