Consider two non zero vectors #v# and #w# in a real linear vector space.
Let #|v|# and #|w|# be the respective norms.
Then construct vectors #A = |w|v + |v|w# and #B = |w|v - |v|w#.
Their inner product is then,
#(A,B) = (|w|v + |v|w,|w|v - |v|w)#
#implies (A,B) = (|w|v,|w|v) - (|w|v,|v|w) + (|v|w,|w|v) - (|v|w,|v|w)#
If the vector space is real, #(|w|v,|v|w) = (|v|w,|w|v)#
#implies (A,B) = |w|^2(v,v) + |v|^2(w,w)#
But, the norm is defined as,
#|v|^2 = (v,v)# and #|w|^2 = (w,w)#
Therefore, #(A,B) = (w,w)(v,v) - (v,v)(w,w)#
Which gives, #(A,B) = 0# that immediately shows that vectors #A = |w|v + |v|w# and #B = |w|v - |v|w# are orthogonal.
In proving this, I have used properties of the inner product. If you are not already familiar with it, check this out :
https://socratic.org/questions/how-is-an-inner-product-defined#472685
