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Chapter 1: Vectors, Lines and Planes
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Section 1.7: Planes
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Essentials


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Definition 1.7.1 gives the equation of a plane in ${\mathrm{\ℝ}}^{3}$, and states the relationship between this equation and the components of a vector perpendicular to the plane.
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Definition 1.7.1

Points $\left(x\,y\,z\right)$ satisfying an equation of the form $axplus;byplus;czplus;dequals;0$ line on a plane and the vector $\mathbf{N}\=a\mathbf{i}plus;b\mathbf{j}plus;c\mathbf{k}$ , called a normal to the plane, is perpendicular to the plane.



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Table 1.7.1 lists the most common ways that a plane can be determined. These six ways are the basis for most exercises found in standard texts on multivariate calculus.
Determining Items

Methodology

Three distinct, noncollinear points

See the four methods detailed in Table 1.7.2.

A point and the normal N

If a plane is to contain point P and have the normal vector N, use either the first or second method listed in Table 1.7.2.

A line and a point not on the line

If a plane is to contain the point P and a line $L$, obtain a normal as follows: Let Q be a point on the line distinct from P, and let the direction of the line be given by the vector V. Calculate $\mathbf{N}\=\mathbf{V}\times \left(\mathbf{P}\mathbf{Q}\right)$ and use either the first or second method listed in Table 1.7.2.

A point and two directions in the plane

If a plane is to contain the point P and the two directions V and W, obtain $\mathbf{N}\=\mathbf{V}\times \mathbf{W}$ and use either the first or second method listed in Table 1.7.2.

Two parallel lines in the plane

If a plane is to contain the parallel lines ${L}_{1}$ and ${L}_{2}$, obtain $\mathbf{N}\={\mathbf{V}}_{1}\times {\mathbf{V}}_{2}$, where ${\mathbf{V}}_{1}$ is the direction of line ${L}_{1}$ and ${\mathbf{V}}_{2}$ is the direction of line ${L}_{2}$. Let P be any point on either line, and use either the first or second method listed in Table 1.7.2.

Two points and a direction in the plane

If a plane is to contain points P and Q, and the direction V, obtain $\mathbf{N}\=\mathbf{V}\times \left(\mathbf{P}\mathbf{Q}\right)$ and use either the first or second method listed in Table 1.7.2.

Table 1.7.1 Determination of a plane



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Table 1.7.2 details different ways to obtain the equation of the plane containing three distinct, noncollinear points ${\mathrm{P}}_{k}$:$\left({x}_{k}\,{y}_{k}\,{z}_{k}\right)\,k\=1\,2\,3$.
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Vector form

$\left(\mathbf{R}\mathbf{}\mathbf{P}\right)\xb7\mathbf{N}\=0$, where $\mathbf{R}\=x\mathbf{i}plus;y\mathbf{j}plus;z\mathbf{k}$ is the position vector to the generic point $\left(x\,y\,z\right)$, P is the position vector to any one of the three given points,
$\mathbf{N}\=\left({\mathbf{P}}_{2}{\mathbf{P}}_{1}\right)\times \left({\mathbf{P}}_{3}{\mathbf{P}}_{1}\right)$
is a normal for the plane, and ${\mathbf{P}}_{k}$ is the position vector to point ${\mathrm{P}}_{k}$.

Adhere to the definition

Obtain $\mathbf{N}\=a\mathbf{i}plus;b\mathbf{j}plus;c\mathbf{k}$ as in the previous cell, then solve for $d$ in an equation of the form $a{x}_{k}plus;b{y}_{k}plus;c{z}_{k}plus;dequals;0$, where $k$ is just one of 1, 2, or 3.

Solve system of equations

Write the set of three equations $a{x}_{k}plus;b{y}_{k}plus;c{z}_{k}plus;dequals;0comma;kequals;1comma;2comma;3$.
Solve for any three of $\left\{a\,b\,c\,d\right\}$ in terms of the fourth; unknowns solved for will appear as linear multiples of that fourth unknown. After substitution back into the equation in Definition 1.7.1, this fourth unknown can then be "divided out" of the resulting equation.

Determinant form

$\left\begin{array}{ccc}x{x}_{1}& y{y}_{1}& z{z}_{1}\\ {x}_{2}{x}_{1}& {y}_{2}{y}_{1}& {z}_{2}{z}_{1}\\ {x}_{3}{x}_{1}& {y}_{3}{y}_{1}& {z}_{3}{z}_{1}\end{array}\right$ = $\left\begin{array}{ccc}x{x}_{1}& y{y}_{1}& z{z}_{1}\\ x{x}_{2}& y{y}_{2}& z{z}_{2}\\ x{x}_{3}& y{y}_{3}& z{z}_{3}\end{array}\right$ = 0

Table 1.7.2 Given three points, obtain the equation of a plane



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•

Skew lines ${L}_{1}$ (with direction ${\mathbf{V}}_{1}$) and ${L}_{2}$ (with direction ${\mathbf{V}}_{2}$) share the common normal $N\={\mathbf{V}}_{1}\times {\mathbf{V}}_{2}$, so the skew lines are contained in the parallel planes given by $\left(\mathbf{R}{\mathbf{P}}_{1}\right)\xb7\mathbf{N}\=0$ and $\left(\mathbf{R}{\mathbf{P}}_{2}\right)\xb7\mathbf{N}\=0$, where R is the position vector to the generic point $\left(x\,y\,z\right)$, and ${P}_{k}\,k\=1\,2$, are points on lines ${L}_{k}\,k\=1\,2$, respectively.

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The distance $s$ from the point P:$\left({x}_{1}\,{y}_{1}\,{z}_{1}\right)$ to the plane $axplus;byplus;czplus;dequals;0$ can be found by a vector approach in which $s$ is the magnitude of the projection of the vector $\mathbf{P}\mathbf{Q}$ on N, the normal for the plane. This approach can also be made to yield the formula${}$

$s\=\frac{\lefta\left({x}_{1}{x}_{0}\right)\+b\left({y}_{1}{y}_{0}\right)\+c\left({z}_{1}{z}_{0}\right)\right}{\sqrt{{a}^{2}\+{b}^{2}\+{c}^{2}}}$ = $\frac{\leftaxplus;byplus;czplus;d\right}{\sqrt{{a}^{2}plus;{b}^{2}plus;{c}^{2}}}$
where Q:$\left({x}_{0}\,{y}_{0}\,{z}_{0}\right)$ is any point on the plane. Likewise, the distance between parallel planes can be found by a projection of $\mathbf{P}\mathbf{Q}$ onto the common normal N, where P is a point in one plane, and Q, a point in the other.
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•

To find the line of intersection of two planes, obtain the direction of the line as the cross product of the normals to the planes, then find one point common to the two planes by calculating the intersection of the two planes and one coordinate plane. The coordinate planes are $x\=c\,y\=c\,z\=c$, where the simplest constant to choose is $c\=0$. Hence, set one variable to zero in each of the two given planes, and solve for the remaining two variables.

•

The angle between two planes is the angle between the normals to the planes.



Examples


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Example 1.7.1

Obtain an equation for the plane containing the points A:$\left(2\,3\,1\right)$, B:$\left(5\,7\,2\right)$, and C:$\left(3\,6\,1\right)$.

Example 1.7.2

Obtain an equation for the plane that passes through the point P:$\left(5\,3\,7\right)$, and has for its normal the vector $\mathbf{N}\=3\mathbf{i}5\mathbf{j}plus;2\mathbf{k}$.

Example 1.7.3

Obtain an equation for the plane that contains the points P:$\left(5\,3\,7\right)$, and Q:$\left(1\,2\,3\right)$, and that is parallel to the vector $\mathbf{V}\=2\mathbf{i}3\mathbf{j}plus;4\mathbf{k}$.

Example 1.7.4

The planes ${S}_{1}\:3x7y9zequals;8$ and ${S}_{2}\:5xplus;4y2zequals;6$ intersect in a line $L$.
a)

Find the parametric equations for $L$.

b)

Find the equation of the plane that is perpendicular to $L$ and that contains the point P:$\left(2\,3\,1\right)$.


Example 1.7.5

Find an equation for ${S}_{3}$, the plane that contains the point P:$\left(2\,3\,1\right)$ and the line of intersection of the planes ${S}_{1}\:3x7y9zequals;8$ and ${S}_{2}\:5xplus;4y2zequals;6$.

Example 1.7.6

Find the distance from the point P:$\left(2\,3\,1\right)$ to the plane $5x7yplus;9zequals;11$.${}$

Example 1.7.7

Find the distance between the parallel planes $2xplus;3yplus;4zequals;5$ and $2xplus;3yplus;4zequals;7$.${}$

Example 1.7.8

Obtain an equation for $\mathrm{\λ}$, the line that passes through the point A:$\left(5\,1\,3\right)$, intersects L, the line $x\=2\+s\,y\=3s1comma;zequals;42s$, and is parallel to P, the plane $3x2yplus;zplus;7equals;0$.

Example 1.7.9

If $P$ is the plane $2xyplus;3zequals;5comma;$obtain an equation for the line $\mathrm{\λ}$ that lies in P, that is perpendicular to $L$, the line given parametrically by
$x\=23tcomma;yequals;1plus;2tcomma;zequals;3plus;t$
and that is at a distance $d\=4$ from M, the point of intersection of $L$ with P.

Example 1.7.10

Obtain an equation for L, the line that is parallel to $P$ and $Q$, the planes
$2x7yplus;9zequals;12$ and $3xplus;5y8zequals;1$
and intersects ${L}_{1}$ and ${L}_{2}$, the lines
$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\+u\left[\begin{array}{c}2\\ 1\\ 1\end{array}\right]$ and $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\=\left[\begin{array}{c}2\\ 1\\ 2\end{array}\right]\+v\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]$

Example 1.7.11

If plane P, and lines ${L}_{1}$ and ${L}_{2}$, are given respectively by
$xy\+z\=6$
$x\=2tplus;1comma;yequals;t1comma;zequals;22t$
$x\=2s\,y\=3\+s\,z\=4\+s$
find an equation for any line $L$ that is parallel to P, and intersects the lines in such a way as to form, between them, a segment of length $3\sqrt{6}$.

Example 1.7.12

Obtain an equation for $Q$, the plane that passes through the point M:$\left(2\,1\,1\right)$, is perpendicular to ${P}_{1}$, the plane $3xplus;yplus;2zequals;28$, and makes an angle of $\mathrm{\π}\/4$ with ${P}_{2}$, the plane $4xplus;3yzequals;1$.

Example 1.7.13

Obtain an equation for P, the plane that contains L, the line
$x\=3\+t\,y\=2\+3tcomma;zequals;4t$
and whose distance from the point A:$\left(2\,1\,3\right)$ is a maximum.

Example 1.7.14

Project point A:$\left(5\,1\,7\right)$ onto $P$, the plane described by the equation $x\+2yplus;3zequals;4$.



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