LemonmathSelf-study sheet

Grade 12 · Lesson 5.3

Angle formulas in space

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Goals

Find the angle between two lines: $\cos\varphi=\dfrac{|\vec{u}_1\cdot\vec{u}_2|}{|\vec{u}_1||\vec{u}_2|}$ .
Find the angle between a line and a plane: $\sin\varphi=\dfrac{|\vec{u}\cdot\vec{n}|}{|\vec{u}||\vec{n}|}$ .
Find the angle between two planes: $\cos\varphi=\dfrac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$ .
Place a cube or a pyramid in $Oxyz$ to compute angles.

Remember

Two lines: $\cos\varphi=\dfrac{|\vec{u}_1\cdot\vec{u}_2|}{|\vec{u}_1||\vec{u}_2|}$ .
A line and a plane: $\sin\varphi=\dfrac{|\vec{u}\cdot\vec{n}|}{|\vec{u}||\vec{n}|}$ (sine, because the angle with the normal is $90^\circ-\varphi$ ).
Two planes: $\cos\varphi=\dfrac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$ .
Every one of these angles lies in $[0^\circ, 90^\circ]$ , thanks to the absolute value.
The ground or a base: $(Oxy)$ with $\vec{k}=(0, 0, 1)$ .
Cubes and pyramids: put the origin at a vertex or at the centre of the base so that many coordinates are $0$ .

Exercises

Exercise 1

Find the angle between $d_1: x=1+t$ , $y=2t$ , $z=-t$ and $d_2: \dfrac{x-1}{2}=\dfrac{y}{-1}=\dfrac{z+3}{1}$ (to one decimal place of a degree).

Exercise 2

Find the angle between the line $d: x=1+t$ , $y=t$ , $z=2$ and the plane $(P): x+z-5=0$ .

Exercise 3

Find the angle between the planes $(P): 2x-y+z-1=0$ and $(Q): x+y+2z+3=0$ .

Exercise 4

Find the angle between the line $\Delta: \dfrac{x-1}{2}=\dfrac{y+1}{-2}=\dfrac{z}{1}$ and the plane $(Oxy)$ (to one decimal place of a degree).

Exercise 5

Let $ABCD.A'B'C'D'$ be a cube with edge $1$ , with $A(0, 0, 0)$ , $B(1, 0, 0)$ , $D(0, 1, 0)$ , $A'(0, 0, 1)$ . Is each claim true or false?

a) The angle between the lines $AC$ and $B'D'$ is $90^\circ$ .

b) The angle between the line $A'C$ and the plane $(ABCD)$ is $45^\circ$ .

c) The angle $\varphi$ between the planes $(A'BD)$ and $(ABCD)$ satisfies $\cos\varphi=\dfrac{1}{\sqrt3}$ .

d) The angle between the lines $AC'$ and $BD$ is $60^\circ$ .

Exercise 6

A wheelchair ramp is an inclined plane: for every $12$ m along the ground it rises $1$ m. Take the ramp as the plane $(P): x-12z=0$ (unit: metre). Find the angle between the ramp and the ground (in degrees, to 1 decimal place).

Exercise 7

The pyramid $S.ABCD$ has a rectangular base with $AB=3$ , $AD=4$ , $SA$ perpendicular to the base and $SA=5$ . Find the angle between $SC$ and the base, and the angle between $SC$ and the plane $(SAB)$ (to one decimal place of a degree).

Exercise 8

(Revisiting The equation of a plane.) What angle does the plane through $A(2, 0, 0)$ , $B(0, 1, 0)$ , $C(0, 0, 2)$ make with the plane $(Oxy)$ (to one decimal place of a degree)?

Exercise 9

(Revisiting The equation of a line in space.) Let $A(1, 2, -1)$ , $B(3, -1, 1)$ and $(P): x-2y+2z+3=0$ . Find the angle between the line $AB$ and $(P)$ (to one decimal place of a degree).

Exercise 10 Challenge

Let $ABCD.A'B'C'D'$ be a cube with edge $1$ . Find the angle between the diagonal $BD'$ and the plane $(A'BD)$ (to one decimal place of a degree).

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. The angle between two lines with direction vectors

\vec{u}_1=(1, 0, 1)

and

\vec{u}_2=(0, -1, -1)

60^\circ

.B.

120^\circ

.C.

30^\circ

.D.

45^\circ

Question 2. The angle between a line with direction vector

\vec{u}=(1, 1, 0)

and a plane with normal vector

\vec{n}=(0, 1, 1)

60^\circ

.B.

45^\circ

.C.

30^\circ

.D.

90^\circ

Question 3. Let

\varphi

be the angle between the plane

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

and the plane

(Oxy)

. Then

\sin\varphi=\dfrac23

.B.

\cos\varphi=\dfrac23

.C.

\cos\varphi=\dfrac29

.D.

\cos\varphi=-\dfrac23

Question 4. In the cube

ABCD.A'B'C'D'

, the angle between the lines

A'C'

and

B'C

45^\circ

.B.

90^\circ

.C.

60^\circ

.D.

120^\circ

Part II. True or false

Question 1. Let

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

and the line

d: x=1+t

y=-2t

z=3

a) The angle between $d$ and $(P)$ is $0^\circ$ .	T / F
b) $(P)$ is perpendicular to the plane $(Q): x+z-4=0$ .	T / F
c) The angle between $(P)$ and $(Oxy)$ is about $41.8^\circ$ .	T / F
d) $d$ is perpendicular to the plane $(Oxy)$ .	T / F

Part III. Short answer

Question 1. Find the angle between the planes

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

and

(Q): 2x+y-z+5=0

, in degrees, to 1 decimal place.

Question 2. A filming drone at

D(0, 0, 40)

points its camera at a target

T(30, 40, 0)

on the ground (unit: metre). Find the angle between the line of sight

DT

and the ground, in degrees, to 1 decimal place.

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

Hints (only when stuck)

Exercise 1.

What you need: $\cos\varphi=\dfrac{|\vec{u}_1\cdot\vec{u}_2|}{|\vec{u}_1||\vec{u}_2|}$ .

Steps:

Read off $\vec{u}_1=(1, 2, -1)$ , $\vec{u}_2=(2, -1, 1)$ .
Compute the scalar product and the two lengths.

Exercise 2.

What you need: $\sin\varphi=\dfrac{|\vec{u}\cdot\vec{n}|}{|\vec{u}||\vec{n}|}$ .

Steps:

$\vec{u}=(1, 1, 0)$ , $\vec{n}=(1, 0, 1)$ .
Compute $\sin\varphi$ .

Exercise 3.

What you need: $\cos\varphi=\dfrac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$ .

Steps:

$\vec{n}_1=(2, -1, 1)$ , $\vec{n}_2=(1, 1, 2)$ .
Compute the cosine.

Exercise 4.

What you need: $(Oxy)$ has normal vector $\vec{k}=(0, 0, 1)$ .

Steps:

$\vec{u}=(2, -2, 1)$ , $|\vec{u}|=3$ .
Compute $\sin\varphi$ .

Exercise 5.

What you need: the three angle formulas; $(A'BD): x+y+z-1=0$ (intercept form).

Steps:

Write the vectors $\overrightarrow{AC}$ , $\overrightarrow{B'D'}$ , $\overrightarrow{A'C}$ , $\overrightarrow{AC'}$ , $\overrightarrow{BD}$ in coordinates.
Use the right formula for each kind of angle.

Exercise 6.

What you need: the angle between two planes; the ground is $(Oxy)$ .

Steps:

$\vec{n}_P=(1, 0, -12)$ , $\vec{k}=(0, 0, 1)$ .
Compute the cosine, then the angle.

Exercise 7.

What you need: put $A$ at the origin; $(ABCD)$ is $(Oxy)$ and $(SAB)$ is $(Ozx): y=0$ .

Steps:

$B(3, 0, 0)$ , $D(0, 4, 0)$ , $S(0, 0, 5)$ , $C(3, 4, 0)$ .
$\overrightarrow{SC}=(3, 4, -5)$ ; use $\sin\varphi$ with $(0, 0, 1)$ and $(0, 1, 0)$ .

Exercise 8.

What you need: the intercept form; the angle between two planes.

Steps:

$(ABC): \dfrac{x}{2}+y+\dfrac{z}{2}=1\iff x+2y+z-2=0$ .
Use $\vec{n}=(1, 2, 1)$ and $\vec{k}=(0, 0, 1)$ .

Exercise 9.

What you need: the line $AB$ has direction vector $\overrightarrow{AB}$ ; $\sin\varphi=\dfrac{|\vec{u}\cdot\vec{n}|}{|\vec{u}||\vec{n}|}$ .

Steps:

$\overrightarrow{AB}=(2, -3, 2)$ .
Compute $\sin\varphi$ with $\vec{n}=(1, -2, 2)$ .

Exercise 10.

What you need: use the axes of exercise 5; $(A'BD): x+y+z-1=0$ .

Steps:

$B(1, 0, 0)$ , $D'(0, 1, 1)$ , so $\overrightarrow{BD'}=(-1, 1, 1)$ .
Compute $\sin\varphi$ with $\vec{n}=(1, 1, 1)$ .

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