4.1 Geometry 203establishes an isomorphism between(R^3 ,×)and(so( 3 ),[·,·]).
Proof.The proof is straightforward if we write the cross-product in coordinates. The
result shows that the cross-product defines a Lie algebra structure on the set of three-
dimensional vectors. Note that the isomorphism maps the sum of vectors to the sum of
matrices, and the dot product of two vectors to the negative of half the trace of the product
of the corresponding matrices.
And now the problems.573.For any three-dimensional vectors−→u,−→v,−→w, prove the identity
−→u×(−→v ×−→w)+−→v ×(−→w×−→u)+−→w×(−→u×−→v)=−→ 0.574.Given three vectors−→a,
−→
b,−→c, define
−→u =(−→b ·−→c)−→a −(−→c ·−→a)−→b,
−→v =(−→a ·−→c)−→b −(−→a ·−→b)−→c,
−→w=(−→b ·−→a)−→c −(−→b ·−→c)−→a.Provethat if−→a,−→
b,−→c form a triangle, then−→u,−→v,−→w also form a triangle, and
this triangle is similar to the first.575.Let−→a,
−→
b,−→c bevectors such that−→
b and−→c are perpendicular, but−→a and−→
b are
not. Letmbe a real number. Solve the system
−→x ·−→a =m,
−→x ×−→b =−→c.576.Consider three linearly independent vectors−→a,
−→
b,−→c in space, having the same
origin. Prove that the plane determined by the endpoints of the vectors is perpen-
dicular to the vector−→a ×−→
b +−→
b ×−→c +−→c ×−→a.577.The vectors−→a,
−→
b, and−→c satisfy
−→a ×−→b =−→b ×−→c =−→c ×−→a =−→ 0.Provethat−→a +−→
b +−→c =−→
0.
578.Find the vector-valued functions−→u(t)satisfying the differential equation
−→u ×−→u′=−→v,where−→v =−→v(t)is a twice-differentiable vector-valued function such that both
−→v and−→v′are never zero or parallel.