Intersection between two lines

• Jun 8th 2010, 11:33 AM
Ulysses
Intersection between two lines
Hi there. I got this exercise, where I must find the intersection between these two lines:

$\displaystyle L_1=2x-y=2$

$\displaystyle L_2=\begin{Bmatrix} x=-1+2\lambda & \mbox{ }& \\y=-1 & \mbox{}& \end{matrix}$

I've parametrized $\displaystyle L_1$

$\displaystyle P_0(1,0)\in{L_1}$ y $\displaystyle P_1(0,-2)\in{L_1}$

From these points of $\displaystyle L_1$ I found a vector parallel to L_1: $\displaystyle u=(-1,-2)$

And then: $\displaystyle L_1=\begin{Bmatrix} x=1-\lambda & \mbox{ }& \\y=-2\lambda & \mbox{}& \end{matrix}$

$\displaystyle L_1\cap{L_2}=\begin{Bmatrix} -1+2\lambda=1-\lambda & \mbox{ }& \\-1=-2\lambda & \mbox{}& \end{matrix}\Longrightarrow{\begin{Bmatrix} \lambda=\displaystyle\frac{3}{2} & \mbox{ }& \\\displaystyle\frac{1}{2}=\lambda & \mbox{}& \end{matrix}}$

Lambda should be in both equations $\displaystyle \displaystyle\frac{1}{2}$, but for x it gives me $\displaystyle \displaystyle\frac{3}{2}$, and I don't know where the error is.

Any help would be appreciated.

Well, I don't know where is the error on my latex code neither :(
• Jun 8th 2010, 12:57 PM
p0oint
Quote:

Originally Posted by Ulysses
Hi there. I got this exercise, where I must find the intersection between these two lines:

$\displaystyle L_1=2x-y=2$

$\displaystyle L_2=\begin{Bmatrix} x=-1+2\lambda & \mbox{ }& \\y=-1 & \mbox{}& \end{matrix}$

I've parametrized $\displaystyle L_1$

$\displaystyle P_0(1,0)\in{L_1}$ y $\displaystyle P_1(0,-2)\in{L_1}$

From these points of $\displaystyle L_1$ I found a vector parallel to L_1: $\displaystyle u=(-1,-2)$

And then: $\displaystyle L_1=\begin{Bmatrix} x=1-\lambda & \mbox{ }& \\y=-2\lambda & \mbox{}& \end{matrix}$

$\displaystyle L_1\cap{L_2}=\begin{Bmatrix} -1+2\lambda=1-\lambda & \mbox{ }& \\-1=-2\lambda & \mbox{}& \end{matrix}\Longrightarrow{\begin{Bmatrix} \lambda=\displaystyle\frac{3}{2} & \mbox{ }& \\\displaystyle\frac{1}{2}=\lambda & \mbox{}& \end{matrix}}$

Lambda should be in both equations $\displaystyle \displaystyle\frac{1}{2}$, but for x it gives me $\displaystyle \displaystyle\frac{3}{2}$, and I don't know where the error is.

Any help would be appreciated.

Well, I don't know where is the error on my latex code neither :(

L1: 2x1-y1=2 or y1=2x1-2

L2: x=-1+2t
y=-1

To find the intersection point you need to do this: y1=y or 2x1-2=-1 and x1=1/2, so the intersection point is (1/2,-1).

Regards.