Atransverse wave is traveling north. in which directions could the wave oscillate?

a transverse wave oscillate in perpendicular direction of its motion.  in the given scenario, the transverse wave is travelling along the north direction. hence the direction of oscillation could be along any of the direction perpendicular to north-south axis.  we know that east-west axis is perpendicular to the north-south axis .  hence the direction of oscillation can be along east-west axis.

we also know that vertically up and down direction is also perpendicular to the north-south axis  hence the direction of oscillation can be along vertically up -down axis.

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west