Question

Do the vectors u = [2 1 -1] v = [1 -1 4] and w [3 2 1] span R3?

Answer 1

Either they span R3 or they do not.

Let's assume they do not, then the 3 vectors must be coplanar, that is
(2,1,-1) = s(1,-1,4) + t(3,2,1)
I will use (..) brackets to stand for vectors for easier typing)

s + 3t = 2
-s + 2t = 1
4s + t = -1

solve the first two equations, add them
5t = 3
t = 3/5 , by subbing back, s = 2/5

sub those values into the third equation:
LS = 4s + t
= 4(2/5) + 3/5
= 11/5
≠ RS
So they vectors can't be coplanar, and therefore must span R3
or they form a basis in R3

Answer 2

vectors u,v,w are coplanar iff u•v×w = 0