Form 4 Chapter 11 Lines and Planes in 3-Dimensions

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


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


MATHEMATICS FORM 4

B. Identify Lines

A B

CD

Lines that lie on a plane

MATHEMATICS FORM 4


Lines that intersect with a plane

A B

CD

MATHEMATICS FORM 4


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


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


Identify the angle between the plane ABCD and the plane BCF.

A B

CD

A B

CD

∠ __ __ __AF B

F

F


Identify the angle between the plane ABC and the plane BCD.

A B

C

A B

C

∠ __ __ __FA B

F

F


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′

Documents

Form 4 Chapter 11 Lines and Planes in 3-Dimensions