Question

Verify that the three points with position vectors u, v, and w lie in the plane:

r.(u x v + v x w + w x u) = [u, v, w]

Answer 1

What's your definition of [u, v, w] ?

Answer 2

I believe it is referring to the scalar triple product. So u . (v x w)

Answer 3

Then isn't this trivial? Substitute r = u, v, w in turn and show that the equation is satisfied.

For example, if r = u,
r.(u x v + v x w + w x u)
= u.u x v + u.v x w + u.w x u
= 0 + [u,v,w] + 0

Now try r = v and r = w

Answer 4

Oh, is that all they're asking for? I guess I just didn't understand the question in the first place :/

Answer 5

...and you need to know that [u,v,w] = [v,w,u] = [w,u,v] = -[u,w,v] = -[w,v,u] = -[v,u,w]