LINES AND PLANES IN
3-DIMENSIONS
CHAPTER 11
MATHEMATICS FORM 4
PRIOR KNOWLEDGE
3 Different types of dimensions3 Different types of dimensions
One surfacelength and width
more than one surface length, width and height
A lineOnly has length
Two- Dimensional Three- DimensionalOne- Dimensional
MATHEMATICS FORM 4
PRIOR KNOWLEDGE
Pythagoras’ TheoremPythagoras’ Theorem
Trigonometric RatiosTrigonometric Ratios
cb
a
MATHEMATICS FORM 4
11.1 ANGLE BETWEEN LINES AND PLANES
A. Identify Plane PLANE: is a flat surface
Plane
Not a Plane
MATHEMATICS FORM 4
11.1 ANGLE BETWEEN LINES AND PLANES
3 types of plane
Vertical plane Inclined planeHorizontal plane
Horizontal plane
Vertical planeVertical plane
Inclined plane
MATHEMATICS FORM 4
Activity 1
1. According to the prism below. Name the specific plane.
A
E
H
D
G
C
B
F Horizontal planeABFEDHGC
Vertical planeABCDEFGHADHE
Inclined planeBFGC
11.1 ANGLE BETWEEN LINES AND PLANES
MATHEMATICS FORM 4
B. Identify Lines
A B
CD
Lines that lie on a plane
MATHEMATICS FORM 4
11.1 ANGLE BETWEEN LINES AND PLANES
Lines that intersect with a plane
A B
CD
MATHEMATICS FORM 4
11.1 ANGLE BETWEEN LINES AND PLANES
Normal to a plane
YP Q
RS
X
Definition: Normal to a plane is a perpendicular straight lineto the intersection of any lines on the plane.
Normal to a plane
MATHEMATICS FORM 4
Activity 2
1. Identify the normal(s) to each of the given planes.
A
E
H
D
G
C
B
F Example:Normal to the plane ADHE are
Answer:AB, DC, EF and HG
MATHEMATICS FORM 4
Activity 2
A
E
H
D
G
C
B
F
(a) Normal to the plane CDHG are
AD and HE
(b) Normal to the plane BCGF are
No normal line
MATHEMATICS FORM 4
11.1 ANGLE BETWEEN LINES AND PLANES
Orthogonal Projection
Definition: Is a perpendicular projection of the object on a plane.
PQ
RS
B
A
Orthogonal projection of line AB on the plane
PQRS
Plane at bottom
Plane at top
Plane at right hand side
Plane at left hand side
Plane at the back
Plane in front
MATHEMATICS FORM 4
REMEMBER THIS…
Imagine …Screen=PLANE Object=LINE
MATHEMATICS FORM 4
Activity 3
1. Find the orthogonal projection of a given line on a specific plane given.
A B
CD
P Q
RS
Line Plane Orthogonal Projection
a) AC ADSP AD
b) BD DCRS CD
c) AR PQRS PR
d) PC ABCD AC
e) QC DCRS RC
f) DQ PQRS SQ
MATHEMATICS FORM 4
Angle between a line and a plane
PQ
RS
B
AOrthogonal projection of line AB on the plane
PQRS is line AC
C
BC is normal to the plane PQRS
Angle between the line AB and the plane PQRS is the angle form between the line AB and the orthogonal projection on the plane.
ANSWER: ∠ B A C
MATHEMATICS FORM 4
TECHNIQUE…∠ __ __ __Point
NOT TOUCH on plane
Point TOUCH on plane
NORMAL of not touch
point on plane
Angle between a line and a plane
PQ
RS
B
A∠ __ __ __Identify the angle between the line AB and the plane PQRS
NOT TOUCH TOUCH NORMAL
C
AB C
Based on the diagram, name the angles between the following:
Answers
Activity 4
NOT TOUCH
TOUCH NORMA
L∠BA
CD
P Q
RS
(a) Line BR and plane ABCD(b) Line AS and plane ABCD
(c) Line AR and plane CDSR(d) Line BS and plane PQRS
∠ __ __ __BR C∠ __ __ __AS D∠ __ __ __RA D∠ __ __ __SB Q
MATHEMATICS FORM 4
Activity 5
1. Identify the angle of the line and the plane given.
A B
CD
P Q
RS
Line Plane Angle
a) AC ADSP ∠CAD
b) BD DCRS ∠BDC
c) AR PQRS ∠ARP
d) PC ABCD ∠PCA
e) QC DCRS ∠QCR
f) DQ PQRS ∠DQS
Based on the diagram, (a)Identify the angle between the
line PB and the plane ABCD.
BA
CD
P Q
RS
3 cm
4 cm
Example 1Example 1
(b) Hence, calculate the angle between the line PB and the plane ABCD.
∠__ __ __BP ANot
TouchTouch
Normal of P
4 cm B
P
A
3cm
tan ∠PBA =∠PBA =tan -1
∠PBA = 36˚52′
A B
CD
P Q
RS(a) Find the angle between the line
SB and the plane ABCD.
(b) Calculate the angle between the line SB and the plane ABCD if SB = 19cm and BD= 13 cm.
D B13 cm
19 cm
S
Example 2Example 2
∠__ __ __BS DNot
TouchTouch
Normal of S
H
A
cos ∠SBD =∠SBD =cos -1
∠SBD= 46˚50′
MATHEMATICS FORM 4
Example 3 (SPM 2006 PAPER 2)Example 3 (SPM 2006 PAPER 2)
Diagram shows a right prism. The base PQRS is a horizontal rectangle. The right angled triangle UPQ is the uniform cross section of the prism.
Identify and calculate the angle between the line RU and the base PQRS. [3 marks]
∠ __ __ __RU P Identify angle
P R
9 cm
U
S
T
U
P Q
R
5 cm12 cm
9 cm
Calculate angle
tan ∠URP=∠URP =tan -1
∠URP= 34˚42′13 cm
MATHEMATICS FORM 4
Example 4(SPM 2008 PAPER 2)Example 4(SPM 2008 PAPER 2)
E
H
A
B
C
GD
F
M
8cm
Diagram shows a cuboid. M is the midpoint of the side EH and AM = 15 cm.
a) Name the angle between the line AM and the plane ADEF
b) Calculate the angle between the line AM and the plane ADEF
[3 marks]
(a) ∠ __ __ __AM E Name angle
(b)
E
A
M
4cm
15 cm
sin ∠MAE=∠MAE=sin -1
∠MAE= 15˚28′H
O
MATHEMATICS FORM 4
T
U
P
Q
R
S
V
5 cm16
cm
12 cm
Example 5 (SPM 2007 PAPER 2)Example 5 (SPM 2007 PAPER 2)
Diagram shows a right prism. The base PQRSIs a horizontal rectangle. Right-angled triangleQRU is the uniform cross section of the prism.V is the midpoint of PS.
Identify and calculate the angle between the line UV and the plane RSTU. [3 marks]
∠ __ __ __UV S
SU
8cm
V
Identify angle
Calculate angle
13 cm
tan ∠VUS=∠VUS= tan -1
∠VUS= 31˚36′
11.2 ANGLE BETWEEN PLANES AND PLANES
Identify the angle between the plane ABCD and the plane BCEF.
A B
CD
E
A B
CD
∠ __ __ __ED C
E
F
F
OR ∠ __ __ __FA B
11.1 ANGLE BETWEEN PLANES AND PLANES
Identify the angle between the plane ABCD and the plane BCF.
A B
CD
A B
CD
∠ __ __ __AF B
F
F
11.1 ANGLE BETWEEN PLANES AND PLANES
Identify the angle between the plane ABC and the plane BCD.
A B
C
A B
C
∠ __ __ __FA B
F
F
11.1 ANGLE BETWEEN PLANES AND PLANES
Identify the angle between the plane ABCD and the plane BCE.
A B
CD
E
A B
CD
E
∠ __ __ __GE F
FG FG
MATHEMATICS FORM 4
SPM 2006 (PAPER 1)
1) Name the angle between the plane PQWT and the plane SRWT.
Q
P
R
S
V
U
W
T
∠ __ __ __SP T
∠ __ __ __RQ W
OR
MATHEMATICS FORM 4
2) Vertex P is vertically above T. Name the angle between the plane PTS and the plane PTQ.
SPM 2007 (PAPER 1)
Q
P
R
ST
∠ __ __ __SQ T
MATHEMATICS FORM 4
3) What is the angle between the plane STU and the base QSTV.
SPM 2008 (PAPER 1)
∠ __ __ __VU T
S
T
U
P
Q
V
MATHEMATICS FORM 4
3) Given M and N is the midpoint of the line QR and PS. Name the angle between the plane VQR and the base PQRS.
SPM 2009 (PAPER 1)
∠ __ __ __NV M
Q
R
U
S
P
V
MN
SPM 2010 (PAPER 2)
MATHEMATICS FORM 4
Diagram in the answer space shows a right prism. The base CDHG is a horizontal rectangle. Trapezium ABCD is the uniform cross section of the prism.
(a)On diagram in the answer space, mark the angle between the plane BCGF and the base CDHG .
(b) Hence, calculate the angle between the plane BCGF and the base CDHG.[3 marks]
Answer :(a)
Mark A
E
H
D
G
C
B
F
21 cm
13 cm
2 cm (b)
X C19 cm
13 cm
B
tan ∠BCD =∠BCD=tan -1
∠BCD= 34˚23′
SPM 2005 (PAPER 2)
MATHEMATICS FORM 4
Diagram shows a right prism. Right-angled triangle PQR is the uniform cross section of the prism.
(a)Name the angle between the plane RTU and the plane PQTU.
(b) Hence, calculate the angle between the plane RTU and the base PQTU.
[3 marks]Answer :(a)
(b)
Q R12 cm
18 cm
Ttan ∠RTQ=
∠RTQ=tan -1
∠RTQ= 33˚41′
∠ __ __ __TR Q
R
P
Q
S
U
T 12 cm5 cm
18 cm
MATHEMATICS FORM 4
SPM 2009 (PAPER 2)
Diagram below shows a cuboid with horizontal base ABCD. J is the midpoint of theside AF.
(a)Name the angle between the plane BCJ and the base ABCD.
(b) Calculate the angle between the plane BCJ and the base ABCD.
[3 marks]
Answer :(a) ∠ __ __ __BJ A
6 cm
8 cm
E
H
A
B
C
GD
F
10 cm
J
(b)
A B10 cm
4 cm
Jtan ∠JBA=
∠JBA=tan -1
∠JBA= 21˚48′
Example (PAPER 2)
∠ __ __ __NV M
Diagram shows a right pyramid. V is the vertex of the pyramid and the base PQRS isa horizontal square. M and N is the midpoint of QR and PS. The height of the pyramid is 11 cm.
Identify and calculate the angle between the plane VQR and the base PQRS.
[3 marks]
Q
R
U
S
P
V
MN
10 cm
5 cm
11 cm
V
MX
tan ∠VMX=∠VMX=tan -1
∠VMX= 65˚33′