Consider three orthogonal vectors in space with direction cosines $l(r), m(r), n(r)$. $(r=1,2,3) $
Show that the determinant of these direction cosines (taken in order) is a constant.
I am not able to find the constant. I worked out the final determinant but am not able to solve it. Need help... thanks in advance!!
Here is the determinant: \begin{vmatrix} l1 & m1 & n1 \\ l2 & m2 & n2 \\ l3 & m3 & n3 \\ \end{vmatrix}
Call $u_r$ the vectors, then the determinant is simply
$$ \Delta = u_1\cdot (u_2 \times u_3) \tag{1} $$
Since the vectors are orthogonal then $u_2 \times u_3 = \alpha u_1$, with $\alpha$ a constant,
$$ \alpha |u_1| = |u_2\times u_3| = |u_2| |u_3| \sin \theta_{23} = \pm|u_2| |u_3| $$
where $\theta_{23}$ is the angle formed by the vectors $u_2$ and $u_3$, which can be either $\pi/2$ or $-\pi/2$. Therefore
$$ \alpha = \pm\frac{|u_2||u_3|}{|u_1|} \tag{2} $$
Evaluating (2) in (1) we then get
$$ \Delta = u_1 \cdot (\alpha u_1) = \alpha |u_1|^2 = \pm|u_1||u_2| |u_3| $$
That is, the determinant depends only on the lengths of the vectors and, therefore, is independent of the direction cosines