Plane

A mathematical plane is a 2-dimensional surface defined using a Vector equation.

Definition §

Vector Equation §

$$\vec{r}={\vec{r}}_0+s\times \vec{u}+t\times \vec{v}$$

• $\vec{r}$ = A vector on the plane
• ${\vec{r}}_0$ = A vector which the plane must pass through
• $\vec{u},\vec{v}$ = Two non-parallel vectors which define the plane
• $s,t$ = Two scalars which are used to parameterise the plane

Point-Normal Form §

We can also define a normal vector, $\vec{n}$ which is always perpendicular to both $\vec{u}$ and $\vec{v}$, making it normal/perpendicular to the plane. We can then use the Dot Product to define the plane:

$${\vec{r}}_0\cdot \vec{n}=\vec{r}\cdot \vec{n}$$

• $\vec{n}$ = Any normal vector perpendicular to $\vec{u}$ and $\vec{v}$. One method to find it is using the Cross Product: $\vec{u}\times \vec{v}$

Cartesian Form §

The cartesian form of a plane can be obtained by converting ${\vec{r}}_0,\vec{r}$ and $\vec{n}$ to their components: $(x_0,y_0,z_0),(x,y,z)$ and $(a,b,c)$:

$$ax+by+cz=d$$

$$d=a{x}_0+b{y}_0+c{z}_0$$

Geometry §

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