Finding the intersection of two lines. What am I doing wrong? Question


*

*Determine whether the pair of lines are parallel and intersection:


$$ r_1 = \langle -1, 0 , 1 \rangle + \lambda \langle 1, 3, 4\rangle$$
$$ r_2 = \langle2, 3, 0\rangle + \mu \langle4, -1, 1\rangle$$
Writing these equations in parametric form gives me:
$$r_1 : $$
$$ x = -1 + \lambda$$
$$ y = 3 \lambda$$
$$ z = 1 + 4 \lambda$$ 
$$r_2 :$$
$$ x = 2 + 4 \mu$$
$$ y = 3 - \mu$$
$$ z = \mu $$
When I solve this equation I get so many solutions that don't make sense. But I don't think these lines are parallel either. Am I doing it wrong?


*

*Two lines have vector equations:


$$ r = 4 \mathbf i + 5 \mathbf j + 6 \mathbf k + t (\mathbf i + 2 \mathbf j + 2 \mathbf k) $$
and
$$ r = -3 \mathbf i + 3 \mathbf j - 8 \mathbf k + t (3 \mathbf i + 2 \mathbf j + 6 \mathbf k) $$
When I move it into parametric form I get:
$$ 4 + 2t = -3 + 3t$$
$$ 5 + 2t = 3 + 2t$$
$$ 6 + 2t = -8 + 6t$$
Which is an unsolvable equation. What am I doing so terribly wrong?
These are two instances where I cannot seem to solve this type of problem and I wonder what I'm doing wrong because this is a 2nd Year question, but I lost all that wiring in my head. 
 A: In three dimensions or higher, a pair of lines could intersection, could be parallel (or coincide, a special case of parallel), or they could be skew.  These two lines appear to be skew.  The easiest way to show this is to try solving the system of three equations in two variables:
$$\begin{eqnarray*}
  -1 + \lambda &=& 2 + 4\mu\\
  3\lambda &=& 3 - \mu \\
  1 + 4\lambda &=& \mu
\end{eqnarray*}
$$
The system is inconsistent (no solution), therefore the lines do not intersect.
And we know the lines are not parallel because the direction vectors $\langle 1, 3, 4 \rangle$ and $\langle 4, -1, 1\rangle$ do not lie in the same line.
A: It's not clear where you're getting hung up, but what you want to do for the first problem is set the equations for $x$, $y$, and $z$ in $\lambda$ and $\mu$ equal to one another, obtaining three equations in two unknowns:
$$\begin{align}
-1+\lambda &= 2+4\mu\cr
3\lambda &= 3-\mu\cr
1+4\lambda &= \mu\cr
\end{align}$$
Typically one does not expect any solutions when there are more equations than unknowns.
As for the second problem, your error there seems to be in thinking that both lines are necessarily described by the same parameter, $t$.  In effect, you're parameterizing the two lines as if each is being traced out by a moving particle, and by setting the parameterizations equal, you're checking whether the particles collide.  But two trajectories can cross without a collision occurring:  think about cars passing at different times through an intersection.  To see that this second problem is similar to the first, change one line's $t$ to $\lambda$ and the other one's $t$ to $\mu$ and then proceed as above.
