Determining whether two given lines intersect or are parallel $$L_1:\left \{\begin{matrix}x & = & 5-6t, \\
y & = & -3t, \\
z & = & -3-9t,\end{matrix}\right .$$$$L_2:\pmatrix{x\\y\\z\\}=\pmatrix{5\\-1\\11\\}+s\pmatrix{3\\-4\\2\\},$$notice that $L_2$ is in parametric form so the equation for $L_2$ would be$$L_2:\left \{\begin{matrix}x & = & 5+3s, \\
y & = & -1-4s, \\
z & = & 11+2s.\end{matrix}\right .$$What is the best way to solve the linear system of equations to see if they intersect?
 A: For the two to intersect, there must be a solution to the overdetermined system
$$5-6t = 5+3s,$$
$$-3t = -1-4s,$$
$$-3-9t = 11+2s.$$
Pick two equations where the coefficients of $s$ and $t$ are not multiples of each other.  If this $2 \times 2$ system has a solution $(s,t)$, substitute into the remaining equation.  If it is satisfied, you have found the intersection.
A: If two lines intersect, they must necessarily lie in the same plane, i.e. be coplanar.
$$L_1: \vec r= (5,0,-3)-\lambda(6,3,9)$$$$=  (5,0,-3)-3\lambda(2,1,3)= (5,0,-3)+\mu(2,1,3)$$ $$L_2: \vec r=(5,-1,11)+\beta(3,-4,2)$$
Now, for $\lambda=0$, $\vec a_1=(5,0,-3)$ lies on $L_1$. Similarly, $\vec a_2=(5,-1,11)$ lies on $L_2$. Then, $\vec a_1-\vec a_2$ lies on the same plane as the lines. However, this must mean that $\vec a_1-\vec a_2=(0,1,-14)$ is coplanar with the vectors $\vec b_1=2\hat i+\hat j+3\hat k$ and $\vec b_2=3\hat i-4\hat j+2\hat k$. Thus, $\vec a_1-\vec a_2$ must be perpendicular to $\vec b_1\times \vec b_2$, i.e. $$(\vec a_1-\vec a_2)\cdot (\vec b_1\times \vec b_2)=0.$$. If the above expression is 0, then the lines intersect, otherwise not.
P.S. The shortest distance between two lines in 3D  space, $L_1:\vec r=\vec a_1+\lambda_1\vec b_1$ and $L_2:\vec r=\vec a_2+\lambda_1\vec b_2$ is $$d_{min}= \frac{(\vec a_1-\vec a_2)\cdot (\vec b_1\times \vec b_2)}{|\vec b_1\times \vec b_2|}.$$ If the lines intersect, the shortest distance is 0.
A: Um, it might be a dumb method, but just assume that they intersect, and let x, y, z be the coordinates that is the intersection point. so you'll have three sets of s-t equations listed as follows :
5 - 6t = 5 + 3s
-3t = -1 -4s
-3 - 9t = 11 + 2s
solve these three equations and check if various s and t sets are the same or not, and you can know if these two lines really intersect or not.
