is sin(x) orthogonal to sin(2x)?

Answer 1

Yes, on a specific interval . . . You can't ignore the space you're in when considering issues of orthogonality and Hermitian properties. On the interval of 0 to 2pi, for instance, these functions are orthogonal.

Answer 2

In the space of functions defined on [0, 2π], where the inner product of f and g is defined to be ∫ f(x)g(x) dx: Yes.

Recall that two objects are orthogonal if their inner product is zero.

∫ (sin (x))(sin (2x)) dx = ∫ (sin (x))(2 sin (x) cos (x)) dx = 2 ∫ sin^2 (x) cos (x) dx.

Now, using the substitution u = sin x, we get

∫ u^2 du, evaluated from u = 0 to u = 0.

This is clearly 0.

Answer 3

I would say no. Graphing this might help.