Inner product that makes vectors an orthonormal basis

Question

Let $X= \begin{pmatrix} a \\ b \end{pmatrix} $ and $Y=\begin{pmatrix} c \\ d \end{pmatrix}$ be two vectors in the plane. Do we have the existence of an inner product that makes $X,Y$ an orthonormal basis?

I think of finding a positive definite matrix $A$ such as $X^TAX=1$ and $X^TAY=0$. Is this the right direction? Does it also work for vectors of higher dimension?

Answer 1

Just define$$(\forall\alpha,\beta,\gamma,\delta\in\mathbb R):\left\langle\alpha X+\beta Y,\gamma X+\delta Y\right\rangle=\alpha\gamma+\beta\delta.$$

Answer 2

If $X$ and $Y$ are linearly dependent you cannot do this. Otherwise you can define $\langle (aX+bY, cX+dY \rangle =ac+bd$.