Homework assignment 2 * Exercise 2.30. Let v ∈ R 3 and v 6 = 0 and consider the linear ODE on R 3 : ˙ x = v × x, where × denotes cross product. Show that the solutions of this ODE are rigid rotations of the initial vector around the direction of the vector v . Writing the ODE as: ˙ x = Sx, show that S =-S T (that is, S is skew-symmetric). Show that the Fow φ t ( x ) = e tS x forms a group of orthogonal transformations. Prove that every solution is periodic and determine the period in terms of v . Solution. The main idea is to think geometrically about this problem, in particular about the geometric interpretation of the cross product of two vectors. Since v × v = 0, it follows that v/ | v | is a unit eigenvector of the matrix S , corresponding to the eigenvalue 0. Choose two orthonormal vectors v ⊥ 1 and v ⊥ 2 in the orthogonal complement of the linear space spanned by v , and such that v/ | v | ,v ⊥ 1 ,v ⊥ 2 (in that order) form a right hand orthonormal basis of R 3 (just like the standard basis
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