Question

What are the different vector and parametric equations for a plane?

Answer 1

A plane in three-dimensional space can be described using different forms of equations, including vector and parametric equations. Here are the main forms:

### 1. Vector Equation of a Plane

A vector equation of a plane can be expressed using a point on the plane and a normal vector to the plane. If \(\mathbf{r}\) is the position vector of a general point \((x, y, z)\) on the plane, \(\mathbf{r_0}\) is the position vector of a specific point \((x_0, y_0, z_0)\) on the plane, and \(\mathbf{n} = \langle a, b, c \rangle\) is a normal vector to the plane, then the vector equation of the plane can be written as:

\[
\mathbf{n} \cdot (\mathbf{r} - \mathbf{r_0}) = 0
\]

This expands to the equation of the plane in the form:

\[
a(x - x_0) + b(y - y_0) + c(z - z_0) = 0
\]

### 2. Scalar (Cartesian) Equation of a Plane

This can be derived from the vector equation. The scalar form of the plane can be written as:

\[
ax + by + cz = d
\]

where \(d = ax_0 + by_0 + cz_0\) for a point \((x_0, y_0, z_0)\) on the plane. This equation gives a simple description of the plane in Cartesian coordinates.

### 3. Parametric Equations of a Plane

To define a plane parametrically, one can use a point on the plane and two non-parallel direction vectors that lie in the plane. Let \(\mathbf{r_0} = \langle x_0, y_0, z_0 \rangle\) be a point on the plane, and let \(\mathbf{d_1} = \langle a_1, b_1, c_1 \rangle\) and \(\mathbf{d_2} = \langle a_2, b_2, c_2 \rangle\) be two direction vectors that lie in the plane. The parametric equations can then be defined as follows:

\[
\mathbf{r}(s, t) = \mathbf{r_0} + s \mathbf{d_1} + t \mathbf{d_2}
\]

where \(s\) and \(t\) are parameters. This can be expressed in components:

\[
\begin{align*}
x &= x_0 + s a_1 + t a_2 \\
y &= y_0 + s b_1 + t b_2 \\
z &= z_0 + s c_1 + t c_2
\end{align*}
\]

### Summary

- **Vector Equation**: \(\mathbf{n} \cdot (\mathbf{r} - \mathbf{r_0}) = 0\)
- **Scalar Equation**: \(ax + by + cz = d\)
- **Parametric Equations**:
\[
\begin{align*}
x &= x_0 + s a_1 + t a_2 \\
y &= y_0 + s b_1 + t b_2 \\
z &= z_0 + s c_1 + t c_2
\end{align*}
\]

These equations and their forms provide various ways to represent a plane in three-dimensional space depending on the context and requirements.