Orthogonal vector space (orthogonal signal sets)

Question

So there are three vectors: $x_1, x_2, x_3$ (which I guess have infinite length?)

on all three axises, respectively

Here are the relevant pages from the textbook: 1, 2

And the tl;dr is that, I have issue with the wording on the text that implies that "g" vector is orthogonal to all three $x_1, x_2$ and $x_3$ vectors.

Which is not the case!

See the picture:

There's no way angle 1, between g and $c_3 x_3$ is 90°, same with angle 2

am I misunderstanding? It's just the text said:

Completeness here means that it is impossible in this space to find any other vector $x_4$, which is orthogonal to all three vectors $x_1, x_2, x_3$.

Which implies that g is orthogonal to $x_1, x_2, x_3$

Answer 1

The textbook states that the three orthogonal vectors are components of g. This implies that a linear combination of these basis vectors is equal to g. Three orthogonal vectors in 3D space span all of 3D space, therefore g cannot be orthogonal to the basis vectors.