What's about the sum of two orthogonal vectors

Question

I'd like to ask if the sum of two orthogonal vectors is a vector which is orthogonal on others, where can I get more details about that?

for example, suppose we have the Walsh matrix of 4, which is

V = [1  1  1  1 
     1 -1  1 -1
     1  1 -1 -1
     1 -1 -1  1]

So V has four orthogonal vector, which are V1 = [1 1 1 1]; V2 = [1 -1 1 -1]; V3 = [1 1 -1 -1] and V4 = [1 -1 -1 1].

My question, if we have V11 = V1 + V2, does it means that V11 is orthogonal on V3, and V4?

where can I get more details about that idea?

thank you

Answer 1

For $u,v,w$ in an inner product space $(V, \langle.,.\rangle)$,

$$\langle v+w, u \rangle =\langle v, u \rangle + \langle w, u \rangle $$

Therefore, if $$\langle v, u \rangle = \langle w, u \rangle = 0$$

then,

$$\langle v+w, u \rangle = 0$$