Vector Equation of a Plane

The vector equation of a plane is a mathematical representation that describes a plane in three-dimensional space using vectors. It can be expressed in the form:

$$\mathbf{r} \cdot \mathbf{n} = \mathbf{r_0} \cdot \mathbf{n}$$

Where:

• $\mathbf{r}$ is the position vector of any point on the plane.
• $\mathbf{n}$ is the normal vector to the plane, which is perpendicular to the plane.
• $\mathbf{r_0}$ is the position vector of a specific point on the plane

The equation states that the dot product of the position vector $\mathbf{r}$ and the normal vector $\mathbf{n}$ is equal to the dot product of a specific point's position vector $\mathbf{r_0}$ and the normal vector $\mathbf{n}$. This means that any point $\mathbf{r}$ that satisfies this equation lies on the plane defined by the normal vector $\mathbf{n}$ and the point $\mathbf{r_0}$.

Note that when expanding the dot products, the equation can also be written in the form:

$$Ax+ By + Cz = 0$$

Where $A$, $B$, and $C$ are the components of the normal vector $\mathbf{n}$, and $x$, $y$, and $z$ are the coordinates of the position vector $\mathbf{r}$. This is the standard form of the equation of a plane in three-dimensional space.