# Dot product for orthonormal basis

I want ask which must be the dot product for vectors (1,1,0), (1,0,1), (0,1,1), so they can form a orthonormalbasis.

Call the $$i$$th such vector $$v_i$$. Let $$V$$ denote the matrix satisfying $$V_{ik}=(v_i)_k$$. We seek an inner product $$a,\,b\mapsto a_kM_{kl}b_l$$, with implicit summation over repeated indices, so that $$\delta_{ij}=V_{ik}M_{kl}V_{jl}$$ i.e. $$VMV^T=I$$. So take $$M=V^{-1}(V^T)^{-1}=(V^TV)^{-1}$$. In this case $$V=\left(\begin{array}{ccc} 1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1 \end{array}\right),\,V^{T}V=\left(\begin{array}{ccc} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end{array}\right),\,M=\frac{1}{4}\left(\begin{array}{ccc} 3 & -1 & -1\\ -1 & 3 & -1\\ -1 & -1 & 3 \end{array}\right).$$
• @melanzana You mean $(3(a_1b_1+a_2b_2+a_3b_3)-a_1b_2-a_2b_1-a_1b_3-a_3b_1-a_2b_3-a_3b_2)/4$? – J.G. Mar 19 at 20:39