Find the equation of the line, $\vec{L}$ where planes ${P_1}$ and ${P_2}$ intersect:

${P_1}$ is given by $2x-3y+5z=5$
${P_2}$ is given by $-2x-3y+5z=3$

1 Answer

Since the line of intersection lies on both planes, the direction $v$ of the line is orthogonal to the normal vectors of both plane. Hence we can take the direction of the line to be
\[v=(2,-3,5)\times (-2,-3,5)=\begin{vmatrix} i & j & k \\ 2 & -3 & 5 \\ -2 & -3 & 5 \end{vmatrix} \]
So $v=-20j-12k$. Next we find a point on the intersection of the two planes. Set $z=0$. Then
\[2x-3y=5,   \text{and}  -2x-3y=3.\]
Solving this system of equations we get 
\[x=\frac{1}{2},  y=-\frac{4}{3}.\]
Thus $(\frac{1}{2}, -\frac{4}{3}, 0 )$ lies on both lines. Hence the parametric equation of the intersection line is
\[x=\frac{1}{2},  y=-\frac{4}{3}-20t,  z= 12 t.\]

  • This type of questions should generally come with a fair bounty. I decided to answer as I thought this might be the first time you are asking a question here :)

    • Thank you so much! and yes this is my first time asking a question.

  • Let me know if you have any questions about the solution.

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