Prove that two lines in R3 intersect and determine the points of intersection

Let

$P = (1,1,1), \qquad Q = (2,1,0).$

Consider the lines $L_1$ and $L_2$ where $L_1$ is the line through $P$ parallel to $A = (1,2,3)$ and $L_2$ is the line through $Q$ parallel to $B = (3,8,13)$ . Prove that these two lines intersect and determine the point of intersection.

Proof. We have

$\begin{align*} L_1 = L(P;A) &= \{ (1+t, 1+2t, 1+3t) \mid t \in \mathbb{R} \} \\ L_2 = L(Q;B) &= \{ (2+3s, 1+8s, 13s) \mid s \in \mathbb{R} \}. \end{align*}$

Then if $L_1$ and $L_2$ intersect we must have $s,t \in \mathbb{R}$ such that

$\begin{align*} 1+t &= 2+3s \\ 1+2t &= 1+8s \\ 1+3t &= 13s. \end{align*}$

From the second equation we have $t - 4s =0$ which implies $t = 4s$ . Plugging this into the first equation we then have $-3s + 4s - 1 = 0$ which implies $s = 1$ . Therefore, $t = 4$ . Then we check these values of $s,t$ satisfy the third equation,