Find the straight line from two plane vectors Two planes have equations $x + 3y - 2z = 4$ and $2x +y +3z = 5$. The planes intersect in the straight line $l$.
Find a vector equation for the line $l$.
How can I do this type of question?

Let $\vec{l}(a,b,c)$.
Hence, $a+3b-2c=0$ and $2a+b+3c=0$, which gives $b=-\frac{7}{11}a$ and $c=-\frac{5}{11}a$.
Let $a=11$. Hence, $b=-7$ and $c=-5$ and $\vec{l}(11,-7,-5)$.
But for $z=0$ we get $x=\frac{11}{5}$ and $y=\frac{3}{5}$, which gets an answer:
$$\left\{\left(\frac{11}{5}+11t,\frac{3}{5}-7t,-5t\right)|t\in\mathbb R\right\}$$
