<s>[INST] Given two lines in Cartesian form, find the vector equation of a line which passes through the intersection of two lines. Given two lines in Cartesian form, find the vector equation of a line which passes through the intersection of two lines (and is perpendicular to both). No points given just the two equations. What are the steps to solve this? 
For example:Line 1: $x+1=\frac{y}{3}=-z$, Line 2: $2x+1=2y+1=z-\frac{3}{2}$.
 [/INST] $x+1 = 2x+1 \Leftrightarrow x = 0$
$\frac{y}{3} = 2y + 1 \Leftrightarrow y = 6y + 3 \Leftrightarrow y = - \frac{3}{5}  $
$ z - \frac{3}{2} = -z \Leftrightarrow 2z = \frac{3}{2} \Leftrightarrow z = \frac{3}{4} $
So the point $P$ of the intersection is : $P=(0,-\frac{3}{5},\frac{3}{4})$
The parallel vectors to the lines are : $n_1 = (1,3,-1) $ and $n_2 = (\frac{1}{2}, \frac{1}{2}, 1)$.
A vector perpendicular to these 2 is : $n_p = n_1 \otimes n_2 = (\frac{7}{2},-\frac{3}{2},-1)  $
$n_p =(\frac{7}{2},-\frac{3}{2},-1)  $  will be the parallel vector of your line and your line passes through $P=(0,-\frac{3}{5},\frac{3}{4})$. 
So the equation of the line is : $\frac{2x-}{7} = -\frac{2y+ \frac{3}{5}}{3}=-z +\frac{3}{4}$
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