I came across a question to prove that the sum of the squares of reciprocals of three mutually perpendicular diameter of an ellipsoid is constant.
To solve this question I assumed ellipse to be $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ and 3 lines are $$\frac{x}{l_1}=\frac{y}{m_1}=\frac{z}{n_1}=1 , \frac{x}{l_2}=\frac{y}{m_2}=\frac{z}{n_2}=1 , \frac{x}{l_3}=\frac{y}{m_3}=\frac{z}{n_3}=1$$
Now i take any arbitrary points on these line like $$(\lambda l_1,\lambda m_1,\lambda n_1)$$ for all lines and find the intersection of line and ellipsoid.
Using this if find the point and length of diameter let's say $$d_1,d_2,d_3$$ and then i calculate $$\frac{1}{d_1^2}+\frac{1}{d_2^2}+\frac{1}{d_2^2} = \frac{1}{4}[\frac{1}{a^2}(l_1^2+l_2^2+l_3^2)+\frac{1}{b^2}(m_1^2+m_2^2+m_3^2)+\frac{1}{c^2}(n_1^2+n_2^2+n_3^2)]$$
Here, when i searched for solution it says that term $$l_1^2+l_2^2+l_3^2=1$$ as lines are perpendicular this is also same for m and n but I know that if lines are perpendiculars then $$l_1l_2+m_1m_2+n_1n_2 = 0$$
Please suggest where things are going wrong at last step.
The idea is to show that if three unit vectors $e_i=(l_i,m_i,n_i)$ are perpendicular to each other, then the matrix $$ U=\begin{pmatrix}l_1&l_2&l_3\\ m_1&m_2&m_3\\ n_1&n_2&n_3\end{pmatrix} $$ is a unitary matrix, (i.e. $UU^T=I$). Then since $U^T$ is also unitary, $l_1^2+l_2^2+l_3^2=1$ and so on.