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powerof
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Homework Statement
We are given a point A(1,1,1) and a vector v=(m-1,3m-5,2m-6). We are asked to write the parametric and continuous (I don't know if that's the appropriate term; in Spanish it's called "forma continua" but you'll see right away what I mean) equations of the line formed by these two.
Then prove that all the possible lines in function of m form a plane.
2. The equations mentioned above
[itex]\left\{\begin{matrix} x=1+(m-1)\lambda
\\ y=1+(3m-5)\lambda
\\ z=1+(2m-6)\lambda
\end{matrix}\right.[/itex]
[itex]\frac{x-1}{m-1}=\frac{y-1}{3m-5}=\frac{z-1}{2m-6}[/itex]
The Attempt at a Solution
I tried to develop from the second equation the other form to write the line, that is, as an intersection of two planes. I wanted to see if the two normal vectors of the planes happened to be linearly dependent but I haven't had any luck so far. You get the following possible pair of planes:
[itex]\left\{\begin{matrix}\pi'\equiv (2m-6)x+(1-m)z+5-m=0
\\ \pi''\equiv(3m-5)x+(1-m)y+4-2m=0
\end{matrix}\right.[/itex]
[itex]\begin{matrix}\overrightarrow{u}_{\pi'}=(2m-6,0,1-m)
\\
\overrightarrow{u}_{\pi''}=(3m-5,1-m,0)
\end{matrix}[/itex]
I honestly don't like this way, it's way too messy in my opinion and I get lost easily. If possible, can I get a few pointers in another direction, to do it another way?