Step 1: Set up coordinates for the unit vector:
Since the direction of a unit vector does not affect the final magnitude relation, choose axes so that $\vec a = (1, 0, 0)$, which is a valid unit vector.
Let $\vec x = (x_1, x_2, x_3)$ be a general vector.
Step 2: Write the dot product in components:
Then $\vec x - \vec a = (x_1 - 1, x_2, x_3)$ and $\vec x + \vec a = (x_1 + 1, x_2, x_3)$.
Their dot product is:
\[ (\vec x-\vec a)\cdot(\vec x+\vec a) = (x_1-1)(x_1+1) + x_2^2 + x_3^2 = x_1^2 - 1 + x_2^2 + x_3^2 \]
Step 3: Recognize the magnitude and solve:
Note that $x_1^2 + x_2^2 + x_3^2 = |\vec x|^2$, so the expression equals $|\vec x|^2 - 1$.
Setting this equal to the given value 12:
\[ |\vec x|^2 - 1 = 12 \quad \Rightarrow \quad |\vec x|^2 = 13 \quad \Rightarrow \quad |\vec x| = \sqrt{13} \]
Final Answer:
The component based calculation confirms the same magnitude for x.
\[ \boxed{|\vec x| = \sqrt{13}} \]