$\mathbf{A}$ and $\mathbf{B}$ are unit vectors on plane $\mathbb{R}^n$ and they are orthogonal to each other. Let $\mathbf{X}$ be the linear combination such that $\mathbf{X} =\mathbf{A}c + \mathbf{B}d$, for some $c, d ∈ R$. Show that $c = \mathbf{X} ·\mathbf{A}$ and $d = \mathbf{X} · \mathbf{B}$
So I tried to construct an equation for dot product: $||X||^2 = (\mathbf{A}c + \mathbf{B}d)·(\mathbf{A}c +\mathbf{B}d)$. But then I got stuck at getting rid of either the $\mathbf{A}$ or $\mathbf{B}$ on either RHS or LHS. I am not sure whether I am on the right track. Can somebody help?
Use the fact that $X.A=(cA+dB).A=c$, since $A.A=1$ and $A.B=0$.