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Angle Between Lines and Planes. Definition. H. E. D. A. G. F. B. C. Identifying Planes. A plane is a flat surface. Examples:. ABCD BCGF CGHD BFEA EFGH. Identify. E. H. The line AC lies on the plane ABCD. F. G. A. D. B. C. Lines on a Plane. - PowerPoint PPT Presentation
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Form 4 Chapter 11: Lines and Planes in 3-Dimensions
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DefinitionA plane is a flat surface.
A
B C
D
E
FG
H ABCD
BCGF
CGHD
BFEA
EFGH
Examples:
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Identify
A
B C
D
E
F G
H
The line AC lies on the plane ABCD.
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Identify
A
B C
D
E
F G
H
The line AH lies on the plane ADHE.
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Identify
A
B C
D
E
F G
H
The line AG intersects with the plane EFCD.
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A line which is perpendicular to any line on the plane that passes through the point of intersection of the line with the plane
Definition
Normal
Plane
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P Q
S R
It is the angle between the line and its orthogonal projection on the plane.
Definition
A
OB
AOB is the angle between the line OA and the plane PQRS.
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Name the angle between the line BH and the plane BCGF.
Example 1
A
B C
D
E
F G
H
Pearson Malaysia Sdn Bhd
Name the angle between the line BH and the plane BCGF.
Example 1
Solution:
A
B C
D
E
F G
H
The line HG is the normal to the plane BCGF.
BG is the orthogonal projection of the line BH on the plane BCGF.
HBG is the angle between the BH and the plane BCGF.
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Name the angle between the line BH and the plane EFGH.
Example 2
A
B C
D
E
F G
H
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Solution:
A
B C
D
E
F G
H
The line BF is the normal to the plane EFGH.
FH is the orthogonal projection of the line BH on the plane EFGH.
BHF is the angle between the BH and the plane EFGH.
Name the angle between the line BH and the plane EFGH.
Example 2
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Name the angle between the line BH and the plane ABFE.
Example 3
A
B C
D
E
F G
H
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Solution:
A
B C
D
E
F G
H
The line EH is the normal to the plane ABFE.
BE is the orthogonal projection of the line BH on the plane ABFE.
HBE is the angle between the BH and the plane ABFE.
Name the angle between the line BH and the plane ABFE.
Example 3
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A
B C
D
E
F G
H
12 cm
10 cm
8 cm
The diagram below shows a model of a cuboid which is made of iron rods. Calculate
Example 4
(a) the length CE,
(b) the angle between the line CE and the plane BCGF.
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Example 4
A
B C
D
E
F G
H
Solution:
(a)
12 cm
10 cm
8 cmB C
F
12 cm
10 cm
In ∆BCF,
CF2 = BC2 + BF2
= 122 + 102
Pythagoras’ theorem
In ∆FCE,
CE2 = CF2 + FE2
= 122 + 102 + 82
FC
E
8 cm
Pythagoras’ theorem
= 308 CE = 17.55 cm
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A
B C
D
E
F G
H
12 cm
10 cm
8 cm
Example 4Solution:
(b)
The angle between the line CE and the plane BCGF is ECF.
FC
E
8 cm
In ∆FCE,
sin ECF =
=
ECF = 27° 7'
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