If OACB is a parallelogrma with `vec( OC) = vec(a) ` and `vec( AB) = vec(b)` then `vec(OA)` is equal to

Options

A. `vec( a) + vec( b)`
B. `vec( a) - vec( b)`
C. `(1)/(2) vec( a) - vec( b)`
D. `(1)/(2) vec( a) - vec( b)`

Answer

To solve the problem, we need to find the vector \(\vec{OA}\) in the parallelogram \(OACB\) given that \(\vec{OC} = \vec{a}\) and \(\vec{AB} = \vec{b}\).

### Step-by-Step Solution:

1. **Understanding the Parallelogram**:
- In a parallelogram, opposite sides are equal and parallel.
- We have points \(O\), \(A\), \(C\), and \(B\) forming the parallelogram \(OACB\).

2. **Assigning the Vectors**:
- Let \(\vec{OC} = \vec{a}\) (given).
- Let \(\vec{AB} = \vec{b}\) (given).
- We need to find \(\vec{OA}\).

3. **Using the Properties of Parallelograms**:
- In a parallelogram, the vector from one vertex to the opposite vertex can be expressed in terms of the other two vectors.
- We can express \(\vec{AC}\) in terms of \(\vec{OA}\) and \(\vec{AB}\):
\[ \vec{AC} = \vec{AB} + \vec{OA} \]

4. **Expressing \(\vec{AC}\)**:
- Since \(\vec{AC}\) is equal to \(\vec{OC}\) (because \(OACB\) is a parallelogram):
\[ \vec{AC} = \vec{OC} = \vec{a} \]

5. **Setting Up the Equation**:
- From the above, we have:
\[ \vec{a} = \vec{b} + \vec{OA} \]

6. **Rearranging the Equation**:
- To find \(\vec{OA}\), we rearrange the equation:
\[ \vec{OA} = \vec{a} - \vec{b} \]

### Final Result:
Thus, the vector \(\vec{OA}\) is given by:
\[ \vec{OA} = \vec{a} - \vec{b} \]