Answer:
Step-by-step explanation:
Suppose x and y are nonzero vectors in an inner product space.
Let us assume that x and y are orthogonal
i.e. innter product is 0.
This implies dot product of x and y is 0
Then x.y =0
i.e. [tex]x^2+y^2 +2x.y = x^2+y^2-2x.y\\||x+y||=||x-y||[/tex]
Proved
Converse part:
Let [tex]||x+y||=||x-y||[/tex]
Square also would be equal
[tex]||x+y||^2=||x-y||^2\\||x||^2+||y||^2+2x,t=||x||^2+||y||^2-2x.y\\x.y =0[/tex]
Hence inner product of x and y is 0
