where e₁ = (n x z) / |n x z| and e₂ = n x e₁ assuming n is a unit vector and z is the unit vector in the z direction of our reference cartesian coordinates.

Now, for those who like arithmetic: what are the cartesian parametric equation of the circle of radius 5.123 centered at point (0.5,1.2,4) and perpendicular to the direction (8,5,6)?

[x,y,z] = [0.5,1.2,4]     (this is the center C)
    + 5.123 cos(θ) [-0.529998940003180, 0.847998304005088, 0]     (this is R cos θ e₁)
    + 5.123 sin(θ) [-0.455083644614378, -0.284427277883986, 0.843800924389160]     (and here's R sin θ e₂)

computed of course, not `by hand', but with this MATLAB code .

Finally, here's a 3D plot of that famous circle

(well, what do you expect, it's just a circle in 3D space!)