How can I prove 3 vectors build orthogonal coordinate system?

Question

I have these 3 vectors. how can I prove that these are built in an orthogonal coordinate system and moreover, determine a normalized base vector??

1 = 1 + 22 − 3

2 = 2 + 23

3 = 51 − 22 + 3

Answer 1

Evaluate $\langle g_i, g_j\rangle$ for all $i,j$ and confirm that $\langle g_i, g_j\rangle = 0$ for $i\neq j$. Then, prove that the $g_i's$ are lienarly independent (i suppose that your vector space has dimension 3). Then, normalize each vector by dividing by its norm.

Answer 2

Hint: Let's begin with 2 vectors first.

2 vectors are orthogonal to eachother if there vectorproduct is 0.