Determine whether the following pair of lines is parallel, intersecting, or skew. If the lines intersect, determine the point(s) of intersection.
$$ r(t) = \langle 4+4t,1-8t,5-3t \rangle $$
$$ R(s) =\langle 14+5s, 5+2s, 13 +4s \rangle $$
I am trying to find an alternative way of resolving this problem using linear algebra. I don't know if it is possible to prove that the lines are parallel or skewed with matrices, if that is possible it would be nice to see how the process looks like. my question is: how i can determine whether these two lines are intersecting.
Best Answer
you have $$ \eqalign{ & {\bf x} = \left( {\matrix{ x \cr y \cr z \cr } } \right) = \left( {\matrix{ 4 \cr 1 \cr 5 \cr } } \right) + t\left( {\matrix{ 4 \cr { - 8} \cr { - 3} \cr } } \right) \cr & {\bf x} = \left( {\matrix{ x \cr y \cr z \cr } } \right) = \left( {\matrix{ {14} \cr 5 \cr {13} \cr } } \right) + s\left( {\matrix{ 5 \cr 2 \cr 4 \cr } } \right) \cr} $$
So the two lines have common points given by the following linear system $$ \left( {\matrix{ 4 \cr 1 \cr 5 \cr } } \right) + t\left( {\matrix{ 4 \cr { - 8} \cr { - 3} \cr } } \right) = \left( {\matrix{ {14} \cr 5 \cr {13} \cr } } \right) + s\left( {\matrix{ 5 \cr 2 \cr 4 \cr } } \right)\quad \Rightarrow \quad \left\{ \matrix{ 4t - 5s = 10 \hfill \cr - 8t - 2s = 4 \hfill \cr - 3t - 4s = 8 \hfill \cr} \right. $$ and you know that this type of systems in general can have $0$, $1$ or infinite solutions according to whether ....