differential equation and linear algebra

1. If vectors in a real inner product space have equal length, then are orthogonal.

Prove it, using the general properties of the inner product.

2. Show that eigenvalues of any real skew-symmetric (matrix are either 0 or pure imaginary.

(Recall the proof of reality for the eigenvalues of a symmetric matrix, modify it slightly)

3. Find the general solution of the system, of course in real form