Prove that v × w ≠ 0 if and only if v and w are linearly independent, and show that || v × w || is the area of the parallelogram with sides v and w.
2.1-5)
Prove that v × w ≠ 0 if and only if v and w are linearly independent, and
show that || v × w || is the area of the parallelogram with sides v and w.
2.1-12)
Angle functions. Let f and g be differentiable real-valued functions on
an interval I. Suppose that f^2 + g2^2= 1 and that J0 is a number such that
f(0) = cos(J0), g(0) = sin(J0.) If J is the function such that
prove that
Hint: We want ( f – cosJ)^2 + (g - sinJ)^2 = 0, so show that its derivative is
zero.
2.6-3)
Find a frame field E1, E2, E3 such that
E1= cos(x)* U1+sin(x) cos(z)* U2 + sin(x) sin(z) * U3
3.2-4)
(a) Prove that an isometry F = TC carries the plane through p orthogonal
to q ≠ 0 to the plane through F(p) orthogonal to C(q).
(b) If P is the plane through (1/2, -1, 0) orthogonal to (0, 1, 0) find an
isometry F = TC such that F(P) is the plane through (1, -2, 1) orthogonal
to (1, 0, -1).
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