1. Use the cross product condition

Given

and if $\alpha$ is the angle between $\vec b$ and $\vec c$, then

So,

2. Find $\vec a \cdot \vec c$

Since

we know that $\vec a$ is perpendicular to both $\vec b$ and $\vec c$. Hence,

3. Compute $|\vec c-\vec a|^2$

Using

and $\vec a\cdot \vec c=0$, we get

Thus,

Therefore,

4. Minimize the expression

We are given

Since $\vec a=\vec b\times\vec c$ and $|\vec a|=2\neq 0$, we must have $\sin\alpha \neq 0$, so effectively

Now, to minimize

we need to maximize $\sin^2\alpha$.

On $\left(0,\frac{\pi}{3}\right]$, the maximum value of $\sin\alpha$ occurs at

So,

Hence,

5. Check options

• A: $124$ ✅
• B: $110$
• C: $121$
• D: $105$

So the correct option is A.