sin(x) as a vector

[Guest]

Hi all!

I need to complete an assignment and I am kinda stuck, so I could do with some help!
This is the info given at the beginning of the assignment:

Fourier series can be seen as a generalisation of the idea of the scalar product. Consider on the one hand the ordinary vectors
v1 = (1, 1) and v2 = (1,°1), and on the other hand the functions w1 = sin(x), w2 = sin(2x), w3 = sin(3x), ..., which we shall think
of as a kind of “vectors” as well.

And this is the question I am at now:
Find the normalised vectors w1* , w2* , w3* , ... (above you may have used Pythagoras’s theorem for this step, but now you will need to formulate the normalisation purely in terms of scalar products, using the usual relation between scalar products and lengths).

For the previous question I indeed used pythagoras to get the normalised vectors for v1 and v2. What I did sofar for this question is the following:

w1∙w1=|w1|^2
sin(x)∙sin(x)=sin(x)^2= |w1|^2
|w1|=sqrt(sin(x)^2)= sin⁡(x)
w1*=w1/|w1| =sin⁡(x)/sin⁡(x)=1

I also got 1 as the answer for w2* and w3*, using the same method. However, the next assignment requires me to check that these vectors are orthogonal so I figured my answers cannot be correct, as they will not be orthogonal if they are all 1.

Can someone please explain to me what I am doing wrong?

[Guest]

I have no idea why you say "they will not be orthogonal if they are all 1". Being "orthogonal" is a matter of the angle between the vectors and has nothing to do with their lengths!

I hope you understand that [tex]|w_1|[/tex] is the length of the vector, not the vector itself.

Now, I have another question- the length of a vector is typically a number, not a vector. But using ordinary function multiplication as your dot product for [tex]w_1[/tex], [tex]w_2[/tex], and [tex]w_3[/tex], you have "sin(x)" for both the vector and its length. Exactly how is your textbook defining the dot product of two functions?
(Typically it is something like [tex]\int_0^{2\pi} f(x)g(x)dx[/tex].)