A long, vertical wire carrying a 10 A current in the upward...

Let the external magnetic field be $\vec{B}_{ext}$ pointing from South to North.

For the resultant magnetic field to be zero, the magnetic field due to the current-carrying wire $\vec{B}_{wire}$ must be equal and opposite to $\vec{B}_{ext}$.

Thus, $\vec{B}_{wire}$ must point from North to South.

Using the right-hand thumb rule, for an upward current, the magnetic field points South on the West side of the wire.

Equating the magnitudes of the magnetic fields:

$B_{wire} = B_{ext}$

$\dfrac{\mu_0 I}{2 \pi r} = 2.0 \times 10^{-3}$

Substituting the given values:

$\dfrac{4 \pi \times 10^{-7} \times 10}{2 \pi r} = 2.0 \times 10^{-3}$

$\dfrac{2 \times 10^{-6}}{r} = 2.0 \times 10^{-3}$

$r = \dfrac{2 \times 10^{-6}}{2.0 \times 10^{-3}}$

$r = 10^{-3} \text{ m} = 1.0 \text{ mm}$

The point is $1.0 \text{ mm}$ West of the wire.