I'm confused about the little n and Big N in this slide.

I believe there is no difference between them.

cos(theta) * u + sin(theta) * (N x u)

What if u is not in plane N? Is u projected onto plane N, then rotated?

will (nxu)x u be u?

(nxu)xu = u?

(nxu)xu should be -n?

N x (N x U) = -U

Rotation by angle theta: cos(theta)U + sin(theta)(NxU)

i think seven is right

n x (n x u) = -u

N x (N x u) = -u

how does the rotation by angle formula work?

n x (n x u) = -u

n x (n x u) = -u

Yes

n x (n x u) = -u

n x (n x u) = -u