# Vector difficulties

• Feb 24th 2009, 06:48 AM
nickskely
Vector difficulties
I am having some difficulties doing some basic vector questions on an assignment.

a line L1 passes through (1,-1,4) and is parallel to vector (1,-1,1). Another line L2 passes through (2,4,7) and (4,5,10). obtain teh vector lines equations for L1 & L2. Do the two lines intersect, if so at what point?

I think I have correctly worked out the line equations as:

for L1: r = (1,-1,4)+
λ(1,-1,1) Edit: thanks to plato
for L2: r = (2,4,7)+
λ(2,1,3)

But I am unsure how to work out if they intersect, and if so where?

Do i need to calculate their cartisan equations, and if so how do I then determine the intersect point.

Thanks for your time
• Feb 24th 2009, 07:13 AM
Plato
Quote:

Originally Posted by nickskely
[I]a line L1 passes through (1,-1,4) and is parallel to vector (1,-1,1). Another line L2 passes through (2,4,7) and (4,5,10). obtain teh vector lines equations for L1 & L2. Do the two lines intersect, if so at what point?
for L2: r = (2,4,7)+[/B]λ(2,1,3)

The first line is $\ell _{1`} :\left\langle {1, - 1,4} \right\rangle + \lambda \left\langle {1, - 1,1} \right\rangle$
• Feb 24th 2009, 07:24 AM
nickskely
Thanks for that Plato, but I am still not sure how to proceed to work out if and where the lines intersect?
• Feb 24th 2009, 08:03 AM
Plato
Quote:

Originally Posted by nickskely
I am still not sure how to proceed to work out if and where the lines intersect?

Write each line in parametric form using a different parameter:
$\ell _1 :\left\{ \begin{gathered}
x = 1 + \lambda \hfill \\
y = - 1 - \lambda \hfill \\
z = 4 + \lambda \hfill \\
\end{gathered} \right.\;\& \,\ell _2 :\left\{ \begin{gathered}
x = 2 + 2\mu \hfill \\
y = 4 + \mu \hfill \\
z = 7 + 3\mu \hfill \\
\end{gathered} \right.$

Now pick two of the coordinates and solve the equations.
(Be sure you check the solution in the coordinate that was not used.)
• Feb 24th 2009, 08:37 AM
nickskely
Thanks, but I am still unclear...

are you saying I need to find the value of λ and μ that will make X (if chosen) equal for both coordiantes?

• Feb 24th 2009, 09:25 AM
Plato
Solve the system: $\begin{gathered} 1 + \lambda = 2 + 2\mu \hfill \\ - 1 - \lambda = 4 + \mu \hfill \\ \end{gathered}$.

But make sure that it also works in $4 + \lambda = z = 7 + 3\mu$.
• Feb 24th 2009, 09:55 AM
nickskely
ok, I see now

So I got:

λ = -3
μ = -2

and it also satisfies Z

So is -3,-2 my intersection point?

and what would the Z value be?
-3,-2,0?

thanks again