For a current element i \, dl located at a position r , the...

According to the Biot-Savart law, the magnetic field $d\vec{B}$ at a position vector $\vec{r}'$ due to a current element $i \, \vec{dl}$ located at a position vector $\vec{r}$ is given by:

$d\vec{B} = \dfrac{\mu_0 i}{4\pi} \dfrac{\vec{dl} \times (\vec{r}' - \vec{r})}{|\vec{r}' - \vec{r}|^3}$

Here, the magnetic field is to be evaluated at the origin, so $\vec{r}' = \vec{0}$.

Substituting $\vec{r}' = \vec{0}$, we get:

$d\vec{B} = \dfrac{\mu_0 i}{4\pi} \dfrac{\vec{dl} \times (\vec{0} - \vec{r})}{|\vec{0} - \vec{r}|^3} = \dfrac{\mu_0 i}{4\pi} \dfrac{\vec{dl} \times (-\vec{r})}{r^3}$

$d\vec{B} = -\dfrac{\mu_0 i}{4\pi} \dfrac{\vec{dl} \times \vec{r}}{r^3}$

Using the property of cross product, $\vec{dl} \times (-\vec{r}) = \vec{r} \times \vec{dl}$.

Therefore, the magnetic field can also be written as:

$d\vec{B} = \dfrac{\mu_0 i}{4\pi} \dfrac{\vec{r} \times \vec{dl}}{r^3}$