Cartesian Coordinates and Cross Product of Vectors for Magnetic Field Direction?

[The black vegetable]

Homework Statement

Homework Equations

The Attempt at a Solution

the answer given is the same but without the negative sign, I don't understand because the crossproduct of unit vectors

when using a Cartesian coordinates of the directions given by the right-hand rule? Is the positive z direction pointing out of the page if X and Y are as follows

apologies if this is in the wrong section, thanks for any help in advance

 

[BvU]

Yes. $\hat {k} = \hat {\bf \imath } \times\hat {\bf \jmath}$ , so z points towards you, out of the screen.

Corkscrew rule I call it. Turn $\hat{\bf \imath }$ over the smallest angle towards $\hat {k}$. Corkscrew will go in the minus y direction : $$\hat \imath \times\hat k = -\hat \jmath $$

 

[TSny]

The problem doesn't specify which of the two wires the charge -q is a distance b from. Does the answer change if you pick the other wire to be a distance b from -q?

 

[The black vegetable]

Thank you both your answers, TSny I thought the same, closer to the top wire the combined magnetic field would be in the opposite direction . Maybe the question is just not very good