Equation of line through a point perpendicular to two other lines?

Question

For point $P(p_1, p_2, p_3)$ and two lines $L_1$ and $L_2$ (with provided Cartesian equations), I assume the new line as $(x-p_1)/l = (y-p_2)/m = (z-p_3)/n$ and using $ll_1 + mm_1 + nn_1 = 0$ and $ll_2 + mm_2 + nn_2 = 0$

get some

$l/a = m/b = n/c$.

Now why is the final solution substituting values of $l,m$ and $n$ with the denominator part?

$(x-p_1)/a = (y-p_2)/b = (z-p_3)/c$

Answer 1

We have:

$L_1: \frac{x-x_1}{l_1}=\frac{y-y_1}{m_2}=\frac {z-z_1}{n_1}$

$L_2: \frac{x-x_2}{l_2}=\frac{y-y_2}{m_2}=\frac {z-z_2}{n_2}$

$a_1=(l_1, m_1, n_1)$.

$a_2=(l_2, m_2, n_2)$

$a=(l, m, n)$

$a=a_1\times a_2$

$${\begin{vmatrix}l&m&n\\l_1&m_1&n_1\\l_2&m_2&n_2\end{vmatrix}}=0$$

Which gives $l$, $m$, $n$ and equation of line $L$ passing through point $P: (p_1,p_2, p_3)$ and perpendicular on lines $L_1$ and $L_2$ is:

$$L: \frac{x-p_1}{l}=\frac{y-p_2}{m}=\frac {z-p_3}{n}$$