A wire oriented in the east-west direction carries a current east -ward.Direction of the magnetic field at a point to the south of wire wil be?

Allright.So draw x-y axis on your paper.The x axis is the wire extending from negative x axis to positive x axis.The current is flowing from negative x axis to positive x axis.We are asked to find out direction of B at any point on negative y axis(or parallel to x axis in south).

Use the right hand screw rule.You will see B is into the plane of the paper,ie,along the negative z axis.

If you want to find it mathematically,use the biosavart law

Here Vec I=vec x and vec l=vec (-y)

The cross product Idl×l will have a direction 

vec x×vec(-y)=-vec z