In sub-parts (i) and (ii) choose the correct option and in sub-parts (iii) to (v), answer the questions as instructed.
Find `abs(vec(x))`, if for a unit vector `vec(a), (vec(x)-vec(a))*(vec(x)+vec(a))=12`

Answer

To solve the problem, we need to find the magnitude of the vector \(\vec{x}\) given the equation:

\[ (\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 12 \]

where \(\vec{a}\) is a unit vector.

### Step-by-Step Solution:

1. **Expand the Dot Product**:
We start by expanding the left-hand side of the equation using the distributive property of the dot product:
\[ (\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = \vec{x} \cdot \vec{x} + \vec{x} \cdot \vec{a} - \vec{a} \cdot \vec{x} - \vec{a} \cdot \vec{a} \]

2. **Simplify the Expression**:
Notice that \(\vec{x} \cdot \vec{a} - \vec{a} \cdot \vec{x} = 0\) (since dot product is commutative). Thus, we can simplify the expression:
\[ \vec{x} \cdot \vec{x} - \vec{a} \cdot \vec{a} = 12 \]

3. **Use the Property of Unit Vector**:
Since \(\vec{a}\) is a unit vector, we know that:
\[ \vec{a} \cdot \vec{a} = 1 \]

4. **Substitute the Value**:
Substitute \(\vec{a} \cdot \vec{a}\) into the equation:
\[ \vec{x} \cdot \vec{x} - 1 = 12 \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ \vec{x} \cdot \vec{x} = 12 + 1 = 13 \]

6. **Find the Magnitude**:
The magnitude of \(\vec{x}\) is given by:
\[ |\vec{x}| = \sqrt{\vec{x} \cdot \vec{x}} = \sqrt{13} \]

### Final Answer:
\[ |\vec{x}| = \sqrt{13} \]