# Find the intersection of a line formed by the intersection of two planes vec(r) vec(n)_1=p_1 and vec(r). vec(n)_2=p_2.

Find the intersection of a line formed by the intersection of two planes $\stackrel{\to }{r}.{\stackrel{\to }{n}}_{1}={p}_{1}$ and $\stackrel{\to }{r}.{\stackrel{\to }{n}}_{2}={p}_{2}$
I know that the line would be along $\left({\stackrel{\to }{n}}_{1}×{\stackrel{\to }{n}}_{2}\right)$. So i need a point on the line to get the equation. I assumed a point C such that $\stackrel{\to }{OC}$ is perpendicular to the line of intersection. I dont really know how to proceed from here. Do I have to use the equations $\stackrel{\to }{c}.{\stackrel{\to }{n}}_{1}={p}_{1}$ and $\stackrel{\to }{c}.{\stackrel{\to }{n}}_{2}={p}_{2}$?
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crearti2d4
You know that the line is of the form $t\left({n}_{1}×{n}_{2}\right)+p$ for some point p. Now to find p, we can assume it is of the form $a{n}_{1}+b{n}_{2}$ (since the line is perpendicular to ${n}_{1}$ and ${n}_{2}$, so it must pass through the plane spanned by them somewhere). Then since p is in both planes, we have the equations $p\cdot {n}_{1}=a+b{n}_{1}\cdot {n}_{2}={p}_{1}$, and $p\cdot {n}_{2}=a{n}_{1}\cdot {n}_{2}+b={p}_{2}$. This gives us the system of equations
$\left(\begin{array}{cc}1& {n}_{1}\cdot {n}_{2}\\ {n}_{1}\cdot {n}_{2}& 1\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}{p}_{1}\\ {p}_{2}\end{array}\right).$
The determinant of the matrix is $1-\left({n}_{1}\cdot {n}_{2}{\right)}^{2}$, and since ${n}_{1}$ and ${n}_{2}$ are not parallel (since the planes intersect in a line, so they are not themselves parallel), this is positive. Hence we can invert the matrix to get
$\left(\begin{array}{c}a\\ b\end{array}\right)=\frac{1}{1-\left({n}_{1}\cdot {n}_{2}{\right)}^{2}}\left(\begin{array}{cc}1& -{n}_{1}\cdot {n}_{2}\\ -{n}_{1}\cdot {n}_{2}& 1\end{array}\right)\left(\begin{array}{c}{p}_{1}\\ {p}_{2}\end{array}\right),$
or letting ${n}_{1}\cdot {n}_{2}=\alpha$
$a=\frac{{p}_{1}-\alpha {p}_{2}}{1-{\alpha }^{2}},$
and
$b=\frac{{p}_{2}-\alpha {p}_{1}}{1-{\alpha }^{2}}.$