Plane through three points

Statement

Write the equation of the plane passing through $A(1;0;0)$, $B(0;2;0)$, $C(0;0;3)$ using the cross product method.

Solution

We construct two vectors in the plane and their cross product as the normal: $\vec{AB}=(-1;2;0), \qquad \vec{AC}=(-1;0;3),$ $\vec n = \vec{AB}\times\vec{AC} = (2\cdot 3-0\cdot 0,\; 0\cdot(-1)-(-1)\cdot 3,\; (-1)\cdot 0-2\cdot(-1)) = (6;3;2).$ Plane through $A$: $6x+3y+2z+d=0$ with $d=-(6\cdot 1)=-6$, that is $\boxed{6x+3y+2z-6=0}.$ Check: at $B$ it gives $6-6=0$, at $C$ it gives $6-6=0$. ✓

Links

Topics: Analytic geometry in space
Concepts: Plane · Cross product · Normal vector
Skills: Analytic geometry · Using formulae
Exercise types: Geometric problem