Perpendicular Vectors in 3D space

Question

I was wondering whether given two Vector's v0 and v1 whether I could find the two perpendicular vectors at a given distance, d, from v1, perpendicular to the v0/v1 line. I know that v0 and v1 will share an axis so the perpendiculars must also.

Example

How would I go about finding two vectors perpendicular to the v0/v1 line, at a right angle either side of v1 at distance d. The perp's would also be at .z = 0.0 in the example shown.

I know that i will always have axis aligned Vectors- but which axes in each case is unknown.

I assume this can be done with the directional vector of the two vectors?

Thanks in advance- hope this makes sense.