Equations of Planes in Three Dimensional Space
We will now look at equations of planes in $\mathbb{R}^3$. There are three forms of planes that we will look at.
| Definition: An equation in the form $Ax + By + Cz + D = 0$ represents the Standard Form Equation of a plane in $\mathbb{R}^3$. |
For example, the equation $2x + 3y + z - 1 = 0$ represents a plane in $\mathbb{R}^3$. The point $(1, 1, -4)$ lies on this plane for example because $(1, 1, -4) \in S = \{ (x, y, z) \in \mathbb{R}^3 : 2x + 3y + z - 1 = 0 \}$ where $S$ is the set of points form $\mathbb{R}^3$ that are contained on this plane.
Now the standard form equation of the plane is the simplest form for planes. We will now look at two other forms a plane can be in, but before we do, we will need the following definition.
| Definition If $\Pi$ is a plane in $\mathbb{R}^3$, then if a vector $\vec{n} = (a, b, c)$ is such that $\vec{n} \perp \Pi$ then $\vec{n}$ is said to be a Normal Vector of The Plane $\Pi$. |
We are now ready to look at the second different form a plane can be in.
Definition: Let $\Pi$ be a plane and let $P_0(x_0, y_0, z_0)$ be any point on $\Pi$, and let $\vec{n} = (a, b, c)$ be a normal vector to $\Pi$. Then the Point-Normal Form Equation of $\Pi$ is $\vec{n} \cdot \vec{P_0P} = 0$.
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We note that the point-normal form equation of the plane consists of all vectors $\vec{P_0P}$ that are perpendicular to the normal $\vec{n}$. Since the point $P_0(x_0, y_0, z_0)$ is on the plane to begin with, the vector $\vec{P_0P}$ is only the plane $\Pi$ only if $P(x, y, z)$ is also on the plane (because if not, then $\vec{n} \cdot \vec{P_0P} \neq 0$).
There is one more form of a plane that we will look at that is similar to the point-normal form.
Definition: Let $\Pi$ be a plane and let $P_0(x_0, y_0, z_0)$ and $P(x, y, z)$ be points on the plane. Let $\vec{r_0}$ be the position vector with terminal point $P_0$ and let $\vec{r}$ be the position vector with terminal point $P$. Then the vector $r - r_0$ is parallel to $\Pi$. Let $\vec{n} = (a, b, c)$ be a normal vector to $\Pi$. The Vector Form Equation of $\Pi$ is $\vec{n} \cdot (\vec{r} - \vec{r_0}) = 0$.
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We will now look at some examples regarding equations of planes in $\mathbb{R}^3$.
Example 1
Determine the equation of the plane that passes through $(1, 1, 1)$ and has the normal vector $\vec{n} = (1, 2, 3)$.
The most convenient form to write this plane in is point-normal form as $(1, 2, 3) \cdot (x - 1, y - 1, z - 1) = 0$. We can expand this equation to get the general form of this plane as follows:
(1)Example 2
Write the point-normal form equation of the plane $2x + 4y + 7z - 2 = 0$.
To write $2x + 4y + 7z - 2 = 0$ in point-normal form we must have a point on the plane and a normal vector to this plane. We can immediately pick up the normal vector for this plane to be $\vec{n} = (2, 4, 7)$. Now we just need a point on the plane. The point $(1, 0, 0)$ is on this plane, and so $(2, 4, 7) \cdot (x - 1, y, z) = 0$ represents the vector-form equation of $2x + 4y + 7z - 2 = 0$.
To verify this, all we need to do is compute the dot product $(2, 4, 7) \cdot (x - 1, y, z)$ as follows:
(2)Example 3
Determine an equation for the plane that passes through $P(6, 2, 3)$, $Q(4, 5, 6)$, and $R(1, 8, 9)$.
We need to find a point on this plane and a normal to this plane. We've been given three points, so all we actually need is a normal vector. To construct a normal vector, consider the vectors $\vec{PQ} = (-2, 3, 3)$ and $\vec{PR} = (-5, 6, 6)$. These vectors both lie on the plane because their initial and terminal points lie on the plane. If we take the cross product of these vectors, we will obtain a vector that is perpendicular to both $\vec{PQ}$ and $\vec{PR}$ and hence a vector that is perpendicular to the plane.
(3)Therefore $\vec{n} = (0, -3, 3)$ will do. So $(0, -3, 3) \cdot (x - 6, y - 2, z - 3) = 0$ represents the plane that passes through $P$, $Q$, and $R$.