Sin(2x) and sin(3x) are orthogonal to each other

[SSSUNNN]

Hi everybody,

I read that sin(2x) and sin(3x) are orthogonal to each other.

In general if I want to check if two functions are orthogonal or not I must integrate their product

First: why the integration of their multiplication (not their addition for example)?

Second: Orthogonal means Perpendicular. But I can't figure out how sin(2x) and sin(3x) are perpendicular

Thank you for your help

 

[Triss]

Orthogonally is a generalization of vectors in $$\vec{v},\vec{w}\in\mathbb{R}^3$$ being perpendicular to each other. You might recall that if $$\vec{v}\cdot\vec{w}= 0$$ then the vectors $$\vec{v}$$ and $$\vec{w}$$ are perpendicular to each other. The dot product is also called an inner product and that space $$\mathbb{R}^3$$ is then called an inner product space.

The set of all continuous functions on the interval $$[a,b]\in\mathbb{R}$$, $$C[a,b]$$ is also an inner product space where the inner product between two functions $$f,g\in C[a,b]$$ is

$$\langle f,g \rangle =\int_a^b f(x)g(x)\,dx.$$

So if you consider the inner product space $$C[-\pi,\pi]$$ then you can see that $$\sin(2x)$$and $$\sin(3x)$$ are orthogonal.

 

[matt grime]

where othogonal means exactly that the inner product is zero (and the vectors are not both zero). it doesn't mean that if you "drew" them they are necessarily at right angles.

 

[HallsofIvy]

The point of $$\langle f,g \rangle =\int_a^b f(x)g(x)\,dx.$$ as "inner product" is that it corresponds to $$\langle(z_1,z_2,z_3),(y_1,y_2,y_3)\rangle= z_1y_1+ z_2y_2+ z_3y_3$$ for 3 dimensional vectors. Think of each x value as a "component" and the integral as summing.

 

[mathwonk]

you use product and not sum because it is a "product", i.e. a bilinear function.