**Linear Algebra**

# Guide on Planes in Linear Algebra

*schedule*Jan 3, 2024

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# Defining a plane using a point and a norm vector

Suppose we are given the following information about some plane:

there exists a point $A$ with coordinates $(x_1,y_1,z_1)$ on the plane.

a vector $\boldsymbol{n}$ that is perpendicular to the plane. This vector is called the norm vector.

Visually, we have the following scenario:

Here, $R$ is another point on the plane with coordinates $(x,y,z)$. The difference between points $A$ and $R$ is that $A$ is assumed to be known, that is, $(x_1,y_1,z_1)$, are some fixed scalar values whereas $R$ can vary.

We know from this sectionlink that the vector pointing from $A$ to $R$, which we denote as $\boldsymbol{v}$, can be expressed as:

Where $\boldsymbol{r}$ and $\boldsymbol{a}$ are the position vectors pointing to $R$ and $A$ respectively.

From theoremlink, because $\boldsymbol{v}$ is perpendicular to the normal vector $\boldsymbol{n}$, their dot product must equal zero:

Let's substitute the vectors into \eqref{eq:NoUqoRBzM8vBpDGtLsK} and simplify:

Here, the right-hand side is some constant because we assume that we are given:

the coordinates of $A$, that is, $(x_1,y_1,z_1)$.

the norm vector $\boldsymbol{n}$, that is, $a$, $b$ and $c$.

We now formally define two important concepts:

the vector equation of a plane.

the cartesian equation of a plane.

# Vector equation of a plane

Suppose we are given that a plane:

passes through the point $A$ represented by a position vector $\boldsymbol{a}$.

has a normal vector $\boldsymbol{n}$.

Let $R$ be some other point on the plane represented by a position vector $\boldsymbol{r}$. The so-called vector equation of a plane is given by:

Where $\boldsymbol{v}$ is a vector pointing from $A$ to $R$ defined as:

Using this, the vector equation of a plane \eqref{eq:YeBAPpdg7dJHKpjIrsU} is also sometimes expressed as:

In some textbooks, the vector equation is equivalently defined as:

Where $\boldsymbol{n}\cdot{\boldsymbol{a}}$ can be thought of as some constant since the components in $\boldsymbol{n}$ and $\boldsymbol{a}$ are assumed to be known.

## Finding the vector equation of a plane

Suppose a plane passes through the point $(2,4,3)$ and has the following normal vector:

Find the vector equation of the plane.

Solution. The plane is parallel to the following vector:

The vector equation of the plane is:

Equivalently, the vector equation of the plane can be expressed as:

If we let vector $\boldsymbol{r}$ be the position vector of any point $(x,y,z)$ on the plane, then the vector equation of the plane can also be compactly expressed as:

We can also compute the dot product to get:

This is called the Cartesian or the general equation of the plane.

# Cartesian (general) equation of the plane

Suppose we are given that a plane passes through the point $(x_1,y_1,z_1)$ and has the normal vector $\boldsymbol{n}$ below:

The Cartesian or the general equation of the plane is:

Where the right-hand side is to be treated as some fixed constant, say $d$.

## Finding the Cartesian equation of a plane given a point and a normal vector

Find the equation of the plane that passes through the point $(1,2,3)$ and has the normal vector below:

Solution. The cartesian equation of the plane is:

## Finding the normal vector of a plane

Consider the following equation of a plane:

Determine a vector that is perpendicular to this plane.

Solution. The normal vector is the coefficients of the cartesian equation of the plane:

Note that any scalar multiple of this normal vector will also be perpendicular to the plane.

# Converting the cartesian form to the vector form

Consider the following general form of a plane:

Determine the vector equation of the plane.

Solution. The normal vector of the plane and the position vector of any point on the plane is:

The vector equation of the plane is:

# Practice problems

Suppose a plane passes through the point $(4,2,5)$ and has the following normal vector:

Find the vector equation of the plane, that is, find $\boldsymbol{n}\cdot{\boldsymbol{r}}$.

We apply the vector equation directly

Find the Cartesian equation of the plane that contains the point $(2,1,3)$ and has the following normal vector:

We can start by finding the vector equation of the plane:

We then expand the dot product on the left-hand side to get the Cartesian equation of the plane: