4 The vector equation of a plane

Consider the plane shown in Figure 49.

Figure 49

Suppose that $A$ is a fixed point in the plane and has position vector $\underset{̲}{a}$ . Suppose that $P$ is any other arbitrary point in the plane with position vector $\underset{̲}{r}$ . Clearly the vector $\overrightarrow{AP}$ lies in the plane.

Task!

Referring to Figure 49, find the vector $\overrightarrow{AP}$ in terms of $\underset{̲}{a}$ and $\underset{̲}{r}$ .

Answer

Also shown in Figure 49 is a vector which is perpendicular to the plane and denoted by $\underset{̲}{n}$ .

Task!

What relationship exists between $\underset{̲}{n}$ and the vector $\overrightarrow{AP}$ ?

Hint: think about the scalar product:

Answer

The answer to the above Task, $\underset{̲}{r}.\underset{̲}{n}=\underset{̲}{a}.\underset{̲}{n},$ is the equation of a plane , written in vector form, passing through $A$ and perpendicular to $\underset{̲}{n}$ .

In this formula it does not matter whether or not $\underset{̲}{n}$ is a unit vector.

If $\underset{̲}{\stackrel{̂}{n}}$ is a unit vector then $\underset{̲}{a}.\underset{̲}{\stackrel{̂}{n}}$ represents the perpendicular distance from the origin to the plane which we usually denote by $d$ (for details of this see Section 9.3). Hence we can write

$\underset{̲}{r}.\underset{̲}{\stackrel{̂}{n}}=d$

This is the equation of a plane , written in vector form, with unit normal $\underset{̲}{\stackrel{̂}{n}}$ and which is a perpendicular distance $d$ from $O$ .

Example 23

1. Find the vector equation of the plane which passes through the point with

   position vector   $3\underset{̲}{i}+2\underset{̲}{j}+5\underset{̲}{k}$ and which is perpendicular to $\underset{̲}{i}+\underset{̲}{k}$ .

2. Find the Cartesian equation of this plane.

Solution

1. Using the previous results we can write down the equation

   $\underset{̲}{r}.\left(\underset{̲}{i}+\underset{̲}{k}\right)=\left(3\underset{̲}{i}+2\underset{̲}{j}+5\underset{̲}{k}\right).\left(\underset{̲}{i}+\underset{̲}{k}\right)=3+5=8$

2. Writing $\underset{̲}{r}$ as $x\underset{̲}{i}+y\underset{̲}{j}+z\underset{̲}{k}$ we have the Cartesian form:

   $\left(x\underset{̲}{i}+y\underset{̲}{j}+z\underset{̲}{k}\right).\left(\underset{̲}{i}+\underset{̲}{k}\right)=8$

   so that

   $x+z=8$

Task!

1. Find the vector equation of the plane through $\left(7,3,-5\right)$ for which

   $\underset{̲}{n}=\left(1,1,1\right)$ is a vector normal to the plane.

2. What is the distance of the plane from $O$ ?

Answer

Exercises

1. Find the equation of a plane which is normal to $8\underset{̲}{i}+9\underset{̲}{j}+\underset{̲}{k}$ and which is a distance 1 from the origin. Give both vector and Cartesian forms.
2. Find the equation of a plane which passes through $\left(8,1,0\right)$ and which is normal to the vector $\underset{̲}{i}+2\underset{̲}{j}-3\underset{̲}{k}$ .
3. What is the distance of the plane $\underset{̲}{r}.\left(\begin{array}{c}\hfill 3\hfill \\ \hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)=5$ from the origin?

Answer