I was going through a proof about a result in spheres (the sum of squares of intercepts of 3 mutually perpendicular lines from the same point to a sphere is constant), and the direction cosines of the three lines were taken as $l_1, m_1, n_1$; $l_2, m_2, n_2$ and $l_3,m_3,n_3$. Two results were used
(i) $l_i^2 + m_i^2 + n_i^2 = 1$, which is clear since they are direction cosines.
(ii) $l_1l_2 + m_1m_2 + n_1n_2 = 0$, and similarly two more equations, which is also clear since they are mutually perpendicular this can be proven using the dot product.
(iii) $l_1m_1 + l_2m_2 + l_3m_3 = 0$, and similarly for $l,n$ and $m,n$ taken together. I am unable to prove this.
I saw that a matrix with the dc's of three mutually perpendicular lines as rows forms an orthogonal matrix, which proves (i), (ii) and (iii), but it is not clear to my why it forms an orthogonal matrix or why (iii) holds. Please help!