What is the value of a if f1(x)=1,f2(x)=x,f3(x)=ax2−1 are orthogonal over [−1,1] ?

Question

How to do this. i got the answers as

1

Answer 1

If orthogonality is defined as <f,g>=∫f(x)g(x)dx=0 where f and g are functions and the interval is on [-1,1], then we can apply this to f1 and f3 and f2 and f3. If f3 is orthogonal to both f1 and f2 then we have orthogonality.

f=f1 and f3=ax²-1: ∫(ax²-1)dx=[ax³/3-x] on [-1,1] = 2a/3-2 = 0 from which a=3.

f=f2: ∫(ax³-x)dx = [ax⁴/4-x²/2] on [-1,1] = 0 for all a because it’s an even function.

So a=3.