Plane

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Plane Equations

$$ax+by+cz=d$$ $$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$

where $⟨x_0,y_0,z_0⟩$ is a point in the plane where $⟨a,b,c⟩=\vec{n}$ is normal to the plane

Determining the Plane

A plane can be determined by

1. 3 points in ${\mathbb{R}}^3$
2. A point & 2 Linearly Independent vectors $||$ to the plane
3. Linear Equation: e.g. $3x-2y+z=4$
4. A point & a normal vector $\vec{n}$

Tip

• The normal vector of a plane and the plane’s equation are bidirectionally derivable
• $✈{\stackrel{◯}{R}}_A||✈{\stackrel{◯}{R}}_B⟺{\vec{n}}_A||{\vec{n}}_B$

See Also

Tangent Plane