Plane

In ${\mathbb{R}}^3$, a plane is spanned by two vectors. We can represent it in standard form:

$$ax+by+cz=d$$ $$a(x-x_0)+b(y-y_0)+c(z-z_0)=d$$ $$d=a{x}_0+b{y}_0+c{z}_0$$

Observe that as a linear equation, planes in ${\mathbb{R}}^3$ have two free variables. We can convert between standard form and a linear combination of vectors fairly trivially this way. We can row reduce to determine the interaction between multiple planes.

Also note that the normal vector that is perpendicular to the plane is given by:

$$\vec{n}=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$$

In ${\mathbb{R}}^n$, a hyperplane is a subspace whose dimension is $n-1$. Note that our definition above still holds!

See also

• Line
• Linear equation
• Tangent plane