Cartesian form of the line

Property — Cartesian form

A line can be described as the intersection of two planes: $\begin{cases}a_1 x+b_1 y+c_1 z+d_1=0\\a_2 x+b_2 y+c_2 z+d_2=0\end{cases}.$ The direction vector of the line is $\vec v = \vec n_1\times\vec n_2$, the cross product of the two normal vectors.

Since the line lies in both planes, its direction is perpendicular to both normal vectors: precisely what the cross product $\vec n_1\times\vec n_2$ provides.

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Topics: Analytic geometry in space
Concepts: Plane · Cross product · Line · Line in space
Skills: Analytic geometry · Using formulae