proof: $l_i: a_i\times (x-x_i)=0$ (i=1, 2) intersects at exactly one point

Question

given: $a_i, x_i\in R^3$, $a_1\times a_2 \neq 0$ and $det(a_1, a_2, x_2-x_1)$=0

prove: $l_i: a_i\times (x-x_i)=0$ (i=1, 2) intersects at exactly one point

Answer 1

$a_i\times (x-x_i)=0$ is a line through $x_i$ in the direction of $a_i$. The condition $det (a_1,a_2,x_2-x_1)=0$ means that $a_1,a_2,x_2-x_1$ lie on a plane, therefore both lines lie on that plane. Since $a_1\times a_2\neq 0$, the lines are not parallel so they intersect at exactly one point.