1. We use the magnitude formula for cross product:

$|\vec a \times \vec b| = |\vec a|\,|\vec b|\sin\theta$

where $\theta$ is the angle between $\vec a$ and $\vec b$.

2. Given:

$|\vec a|=4, \quad |\vec b|=2, \quad \theta=\frac{\pi}{6}$

So,

$|\vec a \times \vec b|=4\cdot 2 \cdot \sin\frac{\pi}{6}$

Since

$\sin\frac{\pi}{6}=\frac{1}{2}$

we get

$|\vec a \times \vec b|=8\cdot \frac{1}{2}=4$

3. Now square it:

$\left(\vec a \times \vec b\right)^2 = |\vec a \times \vec b|^2 = 4^2=16$

4. Check options:

• A: $48$ ❌
• B: $16$ ✅
• C: $\vec a$ ❌
• D: none of these ❌

Hence, the correct answer is B.