Proof of well-known 'Intersection Of Three Planes' formula.
Introduction & Motivation Recently, I found out well-known formula 'Intersection between Three Planes' (Equation (22.31) of [1]) has no proof on literaly anywhere. Even (almost) the book where it is propsed[2]. So, I decided to figure out how this formula is derived.. (of course, experimentally it is wokrs very well.) Notation Let there be three planes on euclidean space $P_1, P_2, P_3$ which intersect in a point. If we let there normals as $n_1=(a_1, b_1, c_1), n_2=(a_2, b_2, c_2), n_3=(a_3, b_3, c_3)$, they must be linearly independent. Also, let the planes $P_1, P_2, P_3$ as the system of equation like below : $$\begin{cases} a_1x+b_1y+c_1z+d_1=0 \\ a_2x+b_2y+c_2z+d_2=0 \\ a_3x+b_3y+c_3z +d_3=0 \end{cases}$$ Proof A. Solve problem with Linear Algebra. To find a intersection point by linear algebra, write system of equations as matrix form : $$\begin{bmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{bmatrix}\begin{bmatrix} x