suppose the u and v are two vectors in R^n. Prove that ||u+v|| = ||u-v|| if and only if u and v are orthogonal. Solution since u and v are orthogonal ? inner product (u,v)=0 Now take L.H.S.= ||u+v||^2 = |u|^2+|v|^2+ 2|(u,v)| =|u|^2+|v|^2 (since (u,v)=0) now R.H.S.= ||u-v||^2 = |u|^2+|v|^2-2|(u,v)| =|u|^2+|v|^2 (since (u,v)=0) ?||u+v||= ||u-v|| conversely suppose that ||u+v||=||u-v|| ?||u+v||^2=||u-v||^2 |u|^2+|v|^2+ 2|(u,v)|=|u|^2+|v|^2-2|(u,v)| ?2|(u,v)|=-2|(u,v)| ?4|(u,v)|=0 ?|(u,v)|=0 ?(u,v)=0 I.e. uand v are orthogonal vectors. = |u|^2+|v|^2-2|(u,v)| =|u|^2+|v|^2 (since (u,v)=0) ?||u+v||= ||u-v|| conversely suppose that ||u+v||=||u-v|| ?||u+v||^2=||u-v||^2 |u|^2+|v|^2+ 2|(u,v)|=|u|^2+|v|^2-2|(u,v)| ?2|(u,v)|=-2|(u,v)| ?4|(u,v)|=0 ?|(u,v)|=0 ?(u,v)=0 I.e. uand v are orthogonal vectors..