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Write a formula for distance from a point to a line. Then count a distance from point $P_1(1,2,4)$ to line $l$, along which intersects the two planes $x+y-2z=1$ and $x+3y-z=4$.

I did a matrix

$$ \left[ \begin{array}{ccc|c} 1&1&-2&1\\ 1&3&1&4\\ \end{array} \right] $$

then transformed it to $$ \left[ \begin{array}{ccc|c} 1&0&-3,5&-0,5\\ 0&1&1,5&1,5\\ \end{array} \right] $$

but completly don't know what to do next.

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  • $\begingroup$ Can you show us what you have tried yourself? $\endgroup$ Feb 16, 2017 at 18:55

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the intersection line of this two planes can we get as follows: mltiplying the first equation by $-1$ and adding to the second we get $$2y+z=3$$ or $$y=\frac{3}{2}-\frac{z}{2}$$ Setting $$z=2t$$ we get the line $$x=-\frac{1}{2}+5t$$ $$y=\frac{3}{2}-t$$ $$z=2t$$ and $t$ is a real number see here for the distance https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line it is the last formula at this page

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