Generate East and North unit vectors tangent to a point on the celestial sphere, using vectors?

Question

I am stuck and a little embarrassed.

I'm trying to implement this answer. I have a unit vector $w$ pointing in the direction of a radio source on the celestial sphere.

$u$ and $v$ are the two other, mutually orthogonal unit vectors that point east and north from that point, tangent to the sphere.

Without back-converting to R.A. and Dec or using trigonometry, is there a simple way to generate $u$ and $v$ vectorially?

Slide 38 here may or may not be helpful, I think my description is sufficient by itself.

Answer 1

If $\mathbf{\hat{n}}$ points to the north celestial pole, then the eastward tangent vector is

$$\mathbf{\hat{u} = \frac{\hat{n} \times \hat{w}}{\|\hat{n} \times \hat{w}\|}}$$

and the northward tangent vector is

$$\mathbf{\hat{v} = \frac{\hat{w} \times \hat{u}}{\|\hat{w} \times \hat{u}\|}}$$

$\mathbf{\|\hat{w} \times \hat{u}\|}$ analytically should be 1 but can be slightly different numerically.