Question

find all unit vectors parallel to the yz-plane that are orthogonal to the vector (3,-1,2)

Answer 1

I'd suggest you first find a vector that is perpendicular (orthogonal) to the given vector <3,-1,2>. The magnitude of your new vector is not important, but the x-, y- and z- components are extremely important.

Answer 2

I haven't done much vector work for 2+ years, but would still suggest the following:

Let the unknown vector be <a, b, c>.

Find the dot product of this vector with <3,-1,2> and set the result equal to zero.
Before you accept this, think about why we'd do that.