I have three equations and have to solve for $x, y, z$.
$$ l_1l_2 + m_1m_2 + n_1n_2 = 0 $$ $$ xl_1 + ym_1 + zn_1 = 0 $$ $$ xl_2 + ym_2 + zn_2 = 0 $$
After eliminating a variable (from the last two), I am left with an equation in two variables. I don't know how to proceed after that - by bringing in the first equation.
This is where I am stuck: $$ y(m_1l_2 - m_2l_1) + z(n_1l_1 - n_2l_1) = 0 $$
I can do the same for other variables too, but that doesn't leave me anywhere - which is obvious since I am not using the first equation. So how to do this?
EDIT: The original problem is to prove that: $$x = m_1n_2 - m_2n_1$$ $$y = n_1l_2 - n_2l_1$$ $$z = l_1m_2 - l_2m_1$$
That can easily done by substituting this in both the equations and showing that it satisfies. But say this wasn't given, what should've been the approach in that case?
Hint: the three equations tell you that the various dot products of three vectors, one of which is $(x,y,z)$, is zero. What other vector can you think of that is orthogonal to two given vectors? There is a big fat hint in the form of the parentheses in the equation you derived.