A long, straight wire conducts a current along the z -axis. It is...

The magnetic field due to a long straight wire carrying current $I$ along the $z$-axis at a point $(x, y)$ in the $x$-$y$ plane is given by $\vec{B} = \dfrac{\mu_0 I}{2 \pi (x^2 + y^2)} (-y \hat{i} + x \hat{j})$.

The magnitude of the magnetic field is $B = \dfrac{\mu_0 I}{2 \pi r}$, where $r = \sqrt{x^2 + y^2}$, and its direction is tangential to the circle of radius $r$ centered at the origin.

For two distinct points to have the same direction of magnetic field, they must lie on the same radial line originating from the origin. For example, points $(x, y)$ and $(2x, 2y)$ have the same direction but different magnitudes. Thus, this is possible.

For two distinct points to have identical magnetic fields, both their magnitudes and directions must be the same. This requires them to be at the same distance and on the same radial line, which means they must be the exact same point. Thus, this is not possible for distinct points.

For two distinct points to have equal magnitudes of magnetic field, they must lie on the same circle centered at the origin. For example, points $(x, y)$ and $(y, x)$ have the same magnitude. Thus, this is possible.

For two distinct points to have opposite magnetic fields, they must lie on the same circle and be diametrically opposite. For example, at points $(x, y)$ and $(-x, -y)$, substituting the coordinates yields vectors exactly opposite to each other. Thus, this is possible.