The magnetic field at the origin due to a current element `I vec(dl)` placed at position `r` is
(i) `((mu_(0)i)/(4pi))((dvec(l)xxvec(r))/(r^(3)))`
`-((mu_(0)i)/(4pi))((dvec(l)xxvec(r))/(r^(3)))`
(iii) `((mu_(0)i)/(4pi))((vec(r)xxdvec(l))/(r^(3)))`
`-((mu_(0)i)/(4pi))((vec(r)xxdvec(l))/(r^(3)))`

Options

A. (i),(ii)
B. (ii),(iii)
C. (i),(ii)
D. (iii),(iv)

Explanation

To solve the problem, we will use the Biot-Savart Law, which describes the magnetic field generated by a current-carrying conductor.

### Step-by-Step Solution:

1. **Understanding the Biot-Savart Law**:
The Biot-Savart Law states that the magnetic field \( \vec{B} \) at a point in space due to a current element \( I \vec{dl} \) is given by:
\[ d\vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{r^3} \]
where \( \vec{r} \) is the position vector from the current element to the point where the magnetic field is being calculated, and \( r \) is the magnitude of \( \vec{r} \).

2. **Positioning the Current Element**:
In this scenario, the current element \( I \vec{dl} \) is located at position \( \vec{r} \) and we want to find the magnetic field at the origin.

3. **Applying the Biot-Savart Law**:
Since we are calculating the magnetic field at the origin due to the current element at position \( \vec{r} \), we can substitute into the Biot-Savart Law:
\[ d\vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{r^3} \]

4. **Considering the Direction**:
The direction of the magnetic field due to a current element follows the right-hand rule. The vector \( d\vec{l} \) is in the direction of the current, and \( \vec{r} \) points from the current element to the origin.

5. **Evaluating the Options**:
- **Option (i)**: \( \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{r^3} \) is incorrect because it does not account for the direction of the magnetic field correctly.
- **Option (ii)**: \( -\frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{r^3} \) is correct as it includes the negative sign indicating the direction of the magnetic field.
- **Option (iii)**: \( \frac{\mu_0 I}{4\pi} \frac{\vec{r} \times d\vec{l}}{r^3} \) is also correct because \( \vec{r} \times d\vec{l} = -d\vec{l} \times \vec{r} \) (due to the properties of the cross product).

6. **Conclusion**:
Therefore, the correct options are (ii) and (iii).

### Final Answer:
The correct options are (ii) and (iii).