Equations of Planes in 3D

Calculus 3 · Axiom Academy

LESSON Equations of Planes in 3D A plane is one point plus one perpendicular direction — the normal vector turns that idea into . 1. A Point and a Normal Fix a Plane Pick a point P_0=(x_0,y_0,z_0) on the plane and a normal vector perpendicular to it. A point lies on the plane exactly when the displacement is perpendicular to — that is, when their dot product is zero. Sweeping through all such perpendicular directions sweeps out the whole flat sheet. The displacement is perpendicular to the normal Expanded component by component: the point-normal form 2. The General Form Reads Off the Normal Multiply out the point-normal equation and collect the constant on the right. The result is the general (scalar) form ax+by+cz=d , where the constant . The payoff: the coefficients of x,y,z are the components of the normal, so you can read a plane's orientation straight off its equation. With P_0=(1,2,3) and , the constant is is the normal . Change them and you tilt the plane. Keep , change d : the plane slides along without rotating. Two planes are parallel exactly when their normals are parallel — same up to a scale. The angle between two planes is just the angle between their normals, from the dot product. Every value of d gives a different member of the same parallel family. As d grows the plane 2x+3y+6z=d marches steadily along the normal — the orientation never changes, only the position. 3. Three Points Build the Normal

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