Step-by-step Solution:

Correct Answer: D (2)

Note on the problem statement: There is a logical inconsistency in the question as written. If $\vec{c} = \vec{a} \times \vec{b}$, then by the definition of the cross product, $\vec{c}$ must be perpendicular to $\vec{a}$, which means their dot product $\vec{a} \cdot \vec{c}$ must be 0. However, the problem states that $\vec{a} \cdot \vec{c} = 2$. This is a contradiction. The problem likely contains a typo, and the intended relation was $\vec{c} = \vec{a} - \vec{b}$. The following solution is based on this correction, as it leads to the provided answer.

Step 1: Assume the Corrected Relation

Let's assume the first condition is $\vec{c} = \vec{a} - \vec{b}$.

The other given conditions are:

• $\vec{a} \cdot \vec{c} = 2$
• $\vec{b} \cdot \vec{c} = 1$
• $|\vec{b}| = 1$

Step 2: Use the Given Conditions to Find $\vec{a} \cdot \vec{b}$

We start with the condition $\vec{b} \cdot \vec{c} = 1$. Substitute $\vec{c} = \vec{a} - \vec{b}$ into this equation:

$\vec{b} \cdot (\vec{a} - \vec{b}) = 1$

Using the distributive property of the dot product:

$\vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} = 1$

$\vec{a} \cdot \vec{b} - |\vec{b}|^2 = 1$

We are given that $|\vec{b}| = 1$, so $|\vec{b}|^2 = 1$. Substitute this value:

$\vec{a} \cdot \vec{b} - 1 = 1 \implies \vec{a} \cdot \vec{b} = 2$

Step 3: Use the Remaining Condition to Find $|\vec{a}|$

Now we use the condition $\vec{a} \cdot \vec{c} = 2$. Again, substitute $\vec{c} = \vec{a} - \vec{b}$:

$\vec{a} \cdot (\vec{a} - \vec{b}) = 2$

$\vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} = 2$

$|\vec{a}|^2 - \vec{a} \cdot \vec{b} = 2$

From Step 2, we found that $\vec{a} \cdot \vec{b} = 2$. Substitute this value:

$|\vec{a}|^2 - 2 = 2$

$|\vec{a}|^2 = 4$

Taking the square root gives the magnitude:

$|\vec{a}| = 2$