LemonmathSelf-study sheet

Grade 12 · Lesson 5.1

The equation of a plane

Name: Date: Time spent: min

Goals

Recognise a normal vector and a pair of direction vectors of a plane.
Write the equation of a plane through a point with a given normal vector, through three points, and in intercept form.
Recognise special planes; decide whether two planes are parallel or perpendicular.
Find the distance from a point to a plane.

Remember

The plane through $M_0(x_0, y_0, z_0)$ with normal vector $\vec{n}=(A, B, C)$ : $A(x-x_0)+B(y-y_0)+C(z-z_0)=0$ .
$Ax+By+Cz+D=0$ has normal vector $(A, B, C)$ .
Direction vectors $\vec{u}=(a, b, c)$ , $\vec{v}=(a', b', c')$ give $\vec{n}=\vec{u}\times\vec{v}=(bc'-b'c,\ ca'-c'a,\ ab'-a'b)$ .
Intercept form: $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$ ; $(Oxy): z=0$ , $(Oyz): x=0$ , $(Ozx): y=0$ .
Parallel: $\vec{n'}=k\vec{n}$ , $D'\neq kD$ . Perpendicular: $\vec{n}\cdot\vec{n'}=0$ .
$d(M_0, (P))=\dfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$ .

Exercises

Exercise 1

Write the equation of the plane through $M(2, 1, -1)$ with normal vector $\vec{n}=(1, -3, 2)$ .

Exercise 2

Let $A(1, 2, 3)$ and $B(3, 0, 5)$ . Write the equation of the perpendicular bisector plane of $AB$ (the plane perpendicular to $AB$ through its midpoint).

Exercise 3

Write the equation of the plane through $A(2, 1, -1)$ , $B(3, 0, 1)$ , $C(1, 3, 0)$ .

Exercise 4

A plane meets the axes $Ox$ , $Oy$ , $Oz$ at $(2, 0, 0)$ , $(0, -3, 0)$ , $(0, 0, 6)$ respectively. Write its Cartesian equation.

Exercise 5

Let $(P): 2x-y+2z-3=0$ . Is each claim true or false?

a) $\vec{n}=(2, -1, 2)$ is a normal vector of $(P)$ .

b) The point $M(1, 2, 1)$ lies in $(P)$ .

c) $(P)$ is parallel to the plane $(Q): 4x-2y+4z+1=0$ .

d) The distance from $O$ to $(P)$ is $3$ .

Exercise 6

A hillside is modelled by the plane $(P): x+2y+2z-30=0$ (unit: metre). A survey drone is at $M(10, 20, 25)$ . Find the distance from the drone to the hillside (in metres, to 1 decimal place).

Exercise 7

Prove that the planes $(P): 2x-2y+z+4=0$ and $(Q): 2x-2y+z-5=0$ are parallel and find the distance between them.

Exercise 8

Write the equation of the plane that contains the axis $Oz$ and passes through $A(1, 2, 5)$ .

Exercise 9

(Revisiting Operations on vectors in coordinates.) In the previous lesson, for $A(1, 0, 2)$ , $B(2, 3, -1)$ , $C(0, 4, 1)$ , we found $D(-1, 1, 4)$ such that $ABCD$ is a parallelogram. Write the equation of the plane $(ABC)$ and check that $D$ lies in it.

Exercise 10 Challenge

Write the equation of the plane $(P)$ that is parallel to $(Q): x+2y+2z+1=0$ and at distance $2$ from the point $A(1, 1, 1)$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. In the system

Oxyz

, a normal vector of the plane

(P): 3x-z+2=0

(3, -1, 2)

.B.

(3, 0, -1)

.C.

(3, -1, 0)

.D.

(0, 3, -1)

Question 2. The plane through

M(1, 2, -1)

with normal vector

\vec{n}=(2, -1, 3)

has equation

2x-y+3z-3=0

.B.

2x-y+3z=0

.C.

2x-y+3z+3=0

.D.

x+2y-z+3=0

Question 3. The distance from

A(1, -2, 3)

to the plane

(P): 2x-y+2z+1=0

\dfrac{10}{3}

.B.

\dfrac{11}{3}

.C.

\dfrac{11}{9}

.D.

11

Question 4. Which plane is parallel to

(P): x-2y+2z-1=0

2x-4y+4z-2=0

.B.

2x+y=0

.C.

-x+2y-2z+3=0

.D.

x+2y+2z-1=0

Part II. True or false

Question 1. In the system

Oxyz

, let

A(2, 0, 0)

B(0, 1, 0)

C(0, 0, 2)

a) The plane $(ABC)$ has equation $x+2y+z-2=0$ .	T / F
b) The point $M(1, 0, 1)$ lies in the plane $(ABC)$ .	T / F
c) The distance from $O$ to $(ABC)$ is $\dfrac13$ .	T / F
d) The plane $(ABC)$ is perpendicular to the plane $(Q): x-y+z+5=0$ .	T / F

Part III. Short answer

Question 1. The plane through

A(1, 2, -1)

parallel to

(Q): 2x+y-2z+7=0

has equation

2x+y-2z+d=0

. Find

d

Question 2. Find the distance between the parallel planes

(P): x+2y-2z+1=0

and

(Q): 2x+4y-4z-7=0

(to 1 decimal place).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $A(x-x_0)+B(y-y_0)+C(z-z_0)=0$ .

Steps:

Substitute $x_0=2$ , $y_0=1$ , $z_0=-1$ and $A=1$ , $B=-3$ , $C=2$ .
Expand and simplify.

Exercise 2.

What you need: the perpendicular bisector plane passes through the midpoint $I$ of $AB$ and has normal vector $\overrightarrow{AB}$ .

Steps:

Find $I$ and $\overrightarrow{AB}$ .
Simplify the normal vector and write the equation.

Exercise 3.

What you need: $\vec{n}=\overrightarrow{AB}\times\overrightarrow{AC}$ is a normal vector of $(ABC)$ .

Steps:

Find $\overrightarrow{AB}$ , $\overrightarrow{AC}$ .
Compute the cross product, then write the equation through $A$ .

Exercise 4.

What you need: the intercept form $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$ .

Steps:

Substitute $a=2$ , $b=-3$ , $c=6$ .
Multiply both sides by $6$ to clear the fractions.

Exercise 5.

What you need: the normal vector $(A, B, C)$ ; the parallel condition $\vec{n'}=k\vec{n}$ , $D'\neq kD$ ; the distance formula.

Steps:

Read off the normal vector and substitute $M$ .
Compare the coefficients of $(P)$ and $(Q)$ .
Compute $d(O, (P))$ .

Exercise 6.

What you need: $d(M, (P))=\dfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$ .

Steps:

Substitute $M$ into the left-hand side and take the absolute value.
Divide by $\sqrt{1+4+4}$ .

Exercise 7.

What you need: the distance between two parallel planes equals the distance from any point of one plane to the other.

Steps:

Compare the normal vectors and the constant terms.
Take an easy point of $(P)$ , for example with $x=y=0$ .

Exercise 8.

What you need: a plane containing $Oz$ passes through $O$ and has $C=0$ , so it has the form $Ax+By=0$ .

Steps:

Substitute $A(1, 2, 5)$ into $Ax+By=0$ .
Choose a pair of numbers $A$ , $B$ that works.

Exercise 9.

What you need: $\vec{n}=\overrightarrow{AB}\times\overrightarrow{AC}$ ; a point lies in a plane when its coordinates satisfy the equation.

Steps:

$\overrightarrow{AB}=(1, 3, -3)$ , $\overrightarrow{AC}=(-1, 4, -1)$ ; compute the cross product.
Write the equation through $A$ , then substitute $D$ .

Exercise 10.

What you need: $(P)$ has the form $x+2y+2z+D=0$ with $D\neq1$ ; the distance formula.

Steps:

Set up $\dfrac{|1+2+2+D|}{3}=2$ .
Solve, then reject any value that makes $(P)$ coincide with $(Q)$ .

Self-check

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

Redo the ones you missed after: ☐ 1 day ☐ 3 days ☐ 7 days

Scan for step-by-step solutions

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐