MHB What are the values of m and n on the line L1?

[Jameson]

The line $L_1$ goes through the point $(4,3,-2)$ and is parallel to the line defined by $(x=1+3t, y=2-4t, z= 3-t)$. If the point $(m,n,-5)$ is also on $L_1$ then find the values of $m$ and $n$.
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[Jameson]

Congratulations to the following members for their correct solutions:

1) MarkFL
2) anemone
3) Sudharaka

Solution (from Sudharaka):

Spoiler

The line \(L_1\) should go through the point \((4,\,3,\,-2)\) and should be parallel to the vector \((3,\,-4,\,-1)\). The equation of the line \(L_1\) in vector form can be written as,

\[L_1:\, (x,\,y,\,z)=(4,\,3,\,-2)+t(3,\,-4,\,-1)\]

Since \((m,\,n,\,-5)\) is on \(L_1\) we have,

\[L_1:\, (m,\,n,\,-5)=(4,\,3,\,-2)+t(3,\,-4,\,-1)\]

\[\Rightarrow t=3\]

\[\therefore m=4+3t=13\mbox{ and }n=3-4t=-9\]