MATHEMATICS
LINES AND PLANES IN
3 DIMENSIONS
FORM 4
LINES AND PLANES IN 3-DIMENSIONS
Introduction
Prior knowledge
Line
Plane
The angle between a line
and a plane
Conclusion
Exit
The angle between
two planes
Prior Knowledge
A.Different types of dimensions
One- Dimensional
Two- Dimensional
B.Pythagoras Theorem
Sine
= Opposite
Hypotenuse
Cosine = Adjacent
Hypotenuse
Tangent = Opposite
Adjacent
Three- Dimensional
PLANES
Definition:
A completely flat surface
HORIZONTAL PLANE
VERTICAL PLANE
INCLINED PLANE
VERTICAL
PLANE
HORIZONTAL
PLANE
INCLINED
PLANE
LINE LIES ON A PLANE
D
Q
P
A
B
C
LINE INTERSECTS WITH A PLANE
X
D
C
A
B
Y
The angle between a line and a plane
Objective: To find and calculate the angle
between a line a plane
Steps to find the angle:
• identify a normal to a plane
• determine the orthogonal projection of the line
• name the angle
• calculate the angle
NORMAL TO A PLANE
Definition:
Normal to a plane is a straight line
which is perpendicular to the intersection
of any lines on the plane.
X
Normal to a plane
S
R
Y
P
Q
Orthogonal projection
Based on the following diagram
The Orthogonal projection is a line which joins
point A with point C. The line lies on the plane
PQRS.
B
BC is normal to
the plane PQRS
S
R
A
P
AC is the orthogonal projection
of line AB on a plane PQRS.
The angle between the line AB and
the plane PQRS is < BAC
C
Q
The angle between a line and a plane
S
Example
R
Q
P
D
A
4 cm
C
3 cm
B
Based on the above diagram, find the angle between the line PB and
the plane ABCD.
The angle between a line and a plane
S
R
P
Q
Name the angle
The angle between the line PB
and the plane ABCD is < ABP
4 cm
D
C
How to calculate the angle ?
A
3 cm
B
How to calculate the angle ?
Find the angle of ABP, if AB = 4 cm, PA = 3 cm
P
S
R
P
Q
3 cm
D
3cm
A
Tan < ABP = AP
AB
=3
4
= 0.75
< ABP = tan –1 0.75
= 36°52 ́
A
4 cm
B
Use a scientific
calculator to find
the answer
C
4 cm
B
The angle between a line and a plane
S
Find the angle between the line SB
and the plane ABCD.
P
Q
Answer :
The angle is < DBS
Calculate the angle of < DBS if SB = 19cm
and BD= 13 cm.
Cos < DBS = DB
SB
= 13
19
= 0.6842
< DBS
R
D
C
A
B
S
19 cm
–1
= cos 0.6842
= 46°49 ́
D
13 cm
B
Exercise
S
P
R
Q
D
A
Based on the diagram,name the angles
between the following:
C
B
(a)Line BR and plane ABCD
(b)Line AS and plane ABCD
(c)Line AR and plane CDSR
(d)Line BS and plane PQRS
Answers
(a) ∠RBC
(b) ∠SAD
(c) ∠ARC
(d) ∠BSQ
E
AF and DE are both perpendicular to the plane
ABCD.AF=DE=5cm, AB=DC=
12cm,AD=BC=9cm,and ABCD is a rectangle.
F
Calculate the angle between
(i) the line BF and the plane ABCD;
(ii)the line BF and the plane ADEF ;
(iii) the line FC and the plane ABCD.
Answers
D
C
5 cm
A
9 cm
12 cm
B
Answers
(i) Since AF is perpendicular to the plane ABCD , the angle between the line
BF and the plane ABCD is given by the angle ABF.
⇒ tan ∠ABF = 5
12
= 0.425
⇒ ∠ABF = 23º
Thus the angle between the line BF and the plane ABCD is 23º
(ii) AF is perpendicular to plane ABCD
⇒ AF is perpendicular to AB.
⇒ The angle between the line BF and the plane ADEF is given by angle ∠AFB
∠AFB = 90º - ∠ABF
= 90º - 23º
= 67º
Thus the angle between the line BF and the plane ADEF is 67º
No(iii)
(iii) Since AF perpendicular to the plane ABCD,the angle between FC and
ABCD is given by the angle ACF.By pythagoras ’ theorem,
AC² = 12² + 9²
⇒ AC = 15cm
in ∆ACF,tan ∠ACF
= AF
AC
= 5
13
⇒ ∠ACF = 21°.
The Angle Between Two Planes
A
B
D
E
C
Intersection Line
F
The angle between two planes which meet
on a line DC is the angle between two
lines, one in each plane,taken
perpendicular to DC and meeting at a
point on DC .
Example 1
Name the angle between planes ADSP and
BDS
Steps to find the angle
Identify the intersection line between
two planes ( ie SD)
ii) Identify perpendicular lines from plane
ADSP and plane BDS to the intersection line
(ie PS/AD and BD respectively)
iii) Identify the perpendicular line which meet
at the same point on the intersection line
(AD and BD meet at the point)
iv) Name the angle
(the angle is ∠ ADB)
S
P
R
Q
i)
D
A
C
B
Example 2
The diagram below shows a cuboid with a rectangular base ABCD.
N is the mid-point of AD. Calculate
a) Find the length of NC
b) Calculate the angle between RN and the base
c) Name the angle between plane RQAN and plane PQRS
6 cm
P
Q
16 cm
Answers
B
A
S
R
N
6 cm
D
C
Back to question
(a) AD = QR = 16 cm
ND = ½AD
= 8 cm
DC = PQ = 6 cm
NC²= 8² + 6²
= 100
NC = √100
= 10 cm
(b) The angle between RN and the base ABCD is
∠RNC.
tan ∠RNC = 6/10
= 0.6
∠RNC = 30°58’
(c) The angle between planes RQAN and
PQRS is ∠AQP.
N
8 cm
D
C
6 cm
R
6 cm
C
N
10 cm
Exercise
EFGH,WXYZ is a cube.Name the angle between;
(i) plane EFYZ and base EFGH;
(ii) plane EFYZ and plane WXYZ;
(iii) plane EFGH and plane HGYZ ;
(iv) plane FHZX and plane HGYZ.
Z
W
H
E
Y
X
F
G
(i) ∠YFG,
(ii) ∠ZEH
∠XYF,
∠WZE
(iii)∠EHZ,
(iv) ∠FGY
∠YZH, ∠FHG
Answers
Multiple Choice Question:
P
E
Q
F
R
S
H
G
Based on the above diagram, which of the following line is not intersect
with plane PQRS?
A FG
B FQ
C HS
D PG
Q
P
R
S
T
U
Based on the above diagram,which of the following line is normal to the plane?
A PF
B PQ
C PS
D ST
P
Q
R
S
E
F
H
G
Based on the above diagram, orthogonal projection for line PG to the
Plane SRGH is
A PS
B RS
C SG
D SH
P
S
Q
R
E
F
H
G
Based on the above diagram, the angle between line SF and the plane
SRGH is
A ∠ FSG
B ∠ FSH
C ∠ SEH
D ∠ SFG
P
S
Q
R
E
F
H
G
The angle between PR and plane PESH is
A 5º43’
B 35º16’
C 45º
D 54º44’
Q
P
S
R
E
H
F
G
Based on the above diagram, the angle between plane PQHG and EFGH is
A ∠ PGE
B ∠ QGF
C ∠ QHF
D ∠ SPH
LINES AND PLANES IN 3DIMENSIONS
LINES
-lies on a plane
-intersect with a plane
PLANES
- Horizontal - inclined
- Vertical
ANGLE BETWEEN A
LINE AND A PLANE
-Normal to plane
-Orthogonal projection
ANGLE BETWEEN TWO
PLANES
-Lines of intersection
-Perpendicular line to the line of
intersection
CALCULATE THE ANGLE
-determine right triangle
-use sine, cosine or tangent to
calculate the angle