# Orthogonal Tensor Proof

1. Sep 30, 2008

### wood0595

I'm curious as to how the following proof is verified. I have toiled over this thing for quite a while, but haven't made any progress. I don't need a step-by-step solution, but I would appreciate any help getting it started:

Given the following:

Q is an orthogonal tensor
e1 is a unit vector such that Qe1 = e1
n is of unit length and orthogonal to e1
m is such that m$$\bullet$$e1 = 0, m = Qn and m$$\bullet$$n= cos$$\theta$$
1 is the identity matrix

Prove that:

Q = 1cos$$\theta$$ + (1-cos$$\theta$$)e1$$\otimes$$e1 - (e2$$\otimes$$e3-e3$$\otimes$$e2)sin$$\theta$$

Where e1, e2, and e3 form an orthonormal triad and
$$\otimes$$ represents the dyadic product

My apologies if this is the wrong thread to post this at, but I looked around and couldn't find any threads on this topic.