Calculate the angles between $(1,X),(X,X^2),(X^2,X^3),(X^3,X^4)$ given the inner product $\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$ Let $V_4$ be the vector space of all polynomials of degree less than or equal to 4 with the inner product $$\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$$ calculate the angles between $(1,X),(X,X^2),(X^2,X^3),(X^3,X^4)$.
My question is are all these polynomials orthogonal to each other? I get $\langle 1,X \rangle =\int_{-1}^{1} XdX=0 , \langle X,X^2 \rangle =\int_{-1}^{1} X^3dX=0, \langle X^2,X^3 \rangle =\int_{-1}^{1} X^5dX=0, \langle X^3,X^4 \rangle = \int_{-1}^{1} X^7dX=0$ 
but I don't think that they are orthogonal to each other.  Can someone please confirm whether they are orthogonal or what I'm doing wrong.


You will note that $PQ\perp AB$ and $AB\perp RS$, yet the three lines are not mutually orthogonal.
