Find the acute angle between the planes with equations
$$
x-2 y+z-9=0 \quad \text { and } \quad x+y-z+2=0
$$
The planes meet in the line $l$, and $A$ is the point on $/$ whose position vector is $p \mathbf{i}+q \mathbf{j}+\mathbf{k}$.
(i) Find $p$ and $q$.
(ii) Find a vector equation for $l$.
Normally to find the intersecting line 3 planes will be provided but here only 2 planes are given, though i have to find 3 variables. please some one help me to find the equation of line of intersection
Best Answer
Writhing down the solutions of the system of equations of the two planes gives you directly their intersection line. e.g. if the planes are $x+y+z=0$ and $x-y-2z=1$, then from the first equation $z=-x-y$. Plug it in the second equation gives $x-y-2(-x-y)=1$, that is $3x+y=1$. Hence the solutions set is $$ \ell=\{(t,1-3t,2t-1):t\in R\}=\{(0,1,-1)+t(1,-3,2):t\in R\} $$ so the parametric equation of the line is $$ \ell(t)=(0,1,-1)+t(1,-3,2) $$