The angle between two vectors `vecaandvecb` with magnitudes `sqrt3` and 4, respectively and `veca.vecb=2sqrt3` is

Options

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

Explanation

To find the angle between the two vectors \(\vec{a}\) and \(\vec{b}\) given their magnitudes and dot product, we can follow these steps:

### Step 1: Write down the given information
We know:
- Magnitude of \(\vec{a} = |\vec{a}| = \sqrt{3}\)
- Magnitude of \(\vec{b} = |\vec{b}| = 4\)
- Dot product \(\vec{a} \cdot \vec{b} = 2\sqrt{3}\)

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

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 2\sqrt{3} = (\sqrt{3}) \cdot (4) \cdot \cos \theta \]

### Step 4: Simplify the equation
Calculating the right side:
\[ 2\sqrt{3} = 4\sqrt{3} \cdot \cos \theta \]
Now, divide both sides by \(4\sqrt{3}\):
\[ \cos \theta = \frac{2\sqrt{3}}{4\sqrt{3}} = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Find the angle \(\theta\)
We know that:
\[ \cos \theta = \frac{1}{2} \]
The angle \(\theta\) that satisfies this equation is:
\[ \theta = \frac{\pi}{3} \text{ radians} \quad \text{(or 60 degrees)} \]

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

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