I have a good geometric intuition of why two non-zero vectors $v_1,v_2$ are orthogonal if $<v_1,v_2>$ = 0 (Where $<,>$ denotes the standard dot product in a euclidean vector space).
But I have read (see source bellow) that if we take a symmetrical bilinear form (hence reflexive), then the same concepts of orthogonality apply: If $s(v_1,v_2) = 0$ for non zero vectors, then these are considered to be orthogonal. I don't see how the concept of orthogonality between objects is preserved when taking symmetric bilinear forms. (Which in my understanding are a generalization of dot products, only keeping the linearity in both components in its most general form)
Any clarification would be greatly appreciated!