The angle between two vectors `vec(a)` and `vec(b)` with magnitudes `sqrt(3)` and 4, respectively and `vec(a). vec(b) = 2 sqrt(3)` is:

Options

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

Answer

To find the angle between the two vectors \(\vec{a}\) and \(\vec{b}\), we can use the formula for the dot product of two vectors, which relates the angle between them to their magnitudes and their dot product.

### Step-by-step Solution:

1. **Identify the given values**:
- Magnitude of \(\vec{a}\): \(|\vec{a}| = \sqrt{3}\)
- Magnitude of \(\vec{b}\): \(|\vec{b}| = 4\)
- Dot product: \(\vec{a} \cdot \vec{b} = 2\sqrt{3}\)

2. **Use the formula for the dot product**:
The dot product of two vectors can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
where \(\theta\) is the angle between the two vectors.

3. **Substitute the known values into the formula**:
\[ 2\sqrt{3} = (\sqrt{3})(4) \cos \theta \]

4. **Simplify the equation**:
\[ 2\sqrt{3} = 4\sqrt{3} \cos \theta \]
Divide both sides by \(\sqrt{3}\):
\[ 2 = 4 \cos \theta \]

5. **Solve for \(\cos \theta\)**:
\[ \cos \theta = \frac{2}{4} = \frac{1}{2} \]

6. **Find the angle \(\theta\)**:
The angle whose cosine is \(\frac{1}{2}\) is:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

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

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