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Q: 6 Let (aij ) be a skew-symmetric 3
× 3 matrix (i.e., aij =
−aji for
all i, j). Let v1,
v2, and v3 be smooth functions of
a parameter
s
satisfying the differential equations
˙
v
i =
3
j
=1
aijvj ,
for i = 1, 2 and 3, and suppose that for some
parameter value s0 the
vectors v1(s0),
v2(s0) and
v3(s0) are orthonormal. Show that the
vectors v1(s),
v2(s), and
v3(s) are orthonormal for all values of
s.
Q2:
Let P be an n×n orthogonal
matrix and let
a ∈ Rn, so that
M
(v
) =
Pv + a is an isometry
of
R3 (see Appendix 1). Show that, if
γ
is a
the unit-speed curve in Rn, the curve
Γ = M(γ) is
also unit-speed. Show
also, if t,
n, b and
T, N,
B are the tangent vector, principal
normal and binormal of γ and
Γ, respectively, then T =
Pt, N =
Pn
and B = Pb.