152012746 Form 4 Chapter 11 Lines and Planes in 3 Dimensions (3)

8/10/2019 152012746 Form 4 Chapter 11 Lines and Planes in 3 Dimensions (3)

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LINES AND PLANES

IN

3-DIMENSIONS

CHAPTER 11


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PRIOR KNOWLEDGE

MATHEMATICS FORM 4

3 Different types of dimensions

One surface

lengthand width

more than one surface

length, widthand height

A line

Only has length

Two- Dimensional Three- DimensionalOne- Dimensional


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11.1 ANGLE BETWEEN LINES AND PLANES

MATHEMATICS FORM 4

A. Identify Plane

PLANE: is aflatsurface

Plane

Not a

Plane


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Horizontal plane

Vertical plane

Vertical plane


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Activity 1

MATHEMATICS FORM 4

1. According to the prism below. Name the specificplane.

A

E

H

DG

C

B

F Horizontal planeABFEDHGC

Vertical planeABCDEFGH

ADHE

Inclined planeBFGC


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11.1 ANGLE BETWEEN LINESAND PLANES

MATHEMATICS FORM 4

B. Identify Lines

A B

CD

Lines that lie on a plane


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MATHEMATICS FORM 4

Lines that intersect with a plane

A B

CD


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MATHEMATICS FORM 4

Normal to a plane

Y

P Q

RS

X

Definition:

Normal to a plane is a perpendicular straight line

to the intersection of any lines on the plane.

Normal to a plane


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Activity 2

MATHEMATICS FORM 4

1. Identify the normal(s) to each of the given planes.

A

E

H

DG

C

B

F Example:

Normal to the plane ADHE areAnswer:AB, DC, EF and HG


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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


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Plane at bottom


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Plane at top


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Plane at right hand side


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Plane at left hand side


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REMEMBER THIS

MATHEMATICS FORM 4

Imagine

Screen=PLANE

Object=LINE


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Activity 3

MATHEMATICS FORM 4

1. Find the orthogonal projection of a given line on aspecific plane given.

A B

CD

P Q

RS

Line Plane OrthogonalProjection

a) AC ADSP AD

b) BD DCRS CD

c) AR PQRS PR

d) PC ABCD AC

e) QC DCRS RC

f) DQ PQRS SQ


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Angle between a line and a plane

MATHEMATICS FORM 4

PQ

RS

B

AOrthogonal projection

of line AB on the plane

PQRS is line AC

C

BC is normal to theplane PQRS

Angle between the line AB and the plane PQRS is the angle form betweenthe line AB and the orthogonal projection on the plane.

ANSWER: B A C


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TECHNIQUE

MATHEMATICS FORM 4

__ __ __Point

NOT TOUCHon plane

PointTOUCHon plane

NORMALof not touch

point on plane

Angle between a line and a plane


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PQ

RS

B

A

__ __ __

Identify the angle between the line ABand the plane PQRS

NOT TOUCH TOUCH NORMAL

C

AB C


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Based on the diagram, name theangles between the following:

Answers

Activity 4

NOTTOUCH

TOUCH NORMAL

BA

CD

P Q

RS

(a) Line BR and plane ABCD

(b) LineAS and planeABCD

(c) Line ARand plane CDSR

(d) Line BSand plane PQRS

__ __ __BR C

__ __ __AS D

__ __ __RA D

__ __ __SB Q


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Activity 5

MATHEMATICS FORM 4

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


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Based on the diagram,

(a)Identify the angle between the

line PBand the plane ABCD.

BA

C

D

P Q

RS

3 cm

4 cm

Example 1

(b) Hence, calculate the anglebetween the line PB and the

plane ABCD.

__ __ __BP ANot

TouchTouch

Normalof P

4 cm B

P

A

3cm

tan PBA =

PBA = tan-1

PBA = 3652


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A B

CD

P Q

RS

(a) Find the angle between the line

SBand 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 2

__ __ __BS DNotTouch

TouchNormal

of S

H

A

cos SBD =

SBD =cos -1

SBD= 4650


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MATHEMATICS FORM 4

Example 3 (SPM 2006 PAPER 2)

Diagram shows a right prism. The basePQRSisa horizontal rectangle. The right angled triangle

UPQis the uniform cross section of the prism.

Identifyand calculatethe angle between theline RU and the base PQRS. [3 marks]

__ __ __RU P Identify angle

P R

9 cm

U

S

T

U

P Q

R

5 cm

9 cm

Calculateangle

tan URP=

URP = tan-1

URP= 3442

13 cm


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MATHEMATICS FORM 4

Example 4(SPM 2008 PAPER 2)

E

H

A

B

C

GD

F

M

8cm

Diagram shows a cuboid.Mis the midpoint ofthe sideEHandAM= 15 cm.

a) Namethe angle between the lineAMandthe planeADEF

b) Calculatethe angle between the lineAMand the planeADEF

[3 marks]

a)

__ __ __AM E Name angle

(b)

E

A

M

4cm

15 cm

sin MAE=

MAE= sin-1

MAE= 1528

H

O


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MATHEMATICS FORM 4

11.2 ANGLE BETWEEN PLANESAND PLANES

Identify the angle between the planeABCDand the planeBCEF.

A B

CD

E

A B

CD

__ __ __ED C

E

F

F

R

__ __ __FA B


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MATHEMATICS FORM 4


Identify the angle between the planeABCDand the planeBCF.

A B

CD

A B

CD

__ __ __AF B

F

F


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MATHEMATICS FORM 4


Identify the angle between the planeABCand the planeBCD.

A B

C

A B

C

__ __ __FA B

F

F


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SPM 2006 (PAPER 1)

MATHEMATICS FORM 4

1) Name the angle between the planePQWTand theplaneSRWT.

Q

P

R

S

V

U

W

T

__ __ __SP T

__ __ __RQ W

OR


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MATHEMATICS FORM 4

3) What is the angle between the plane STU and thebase QSTV.

SPM 2008 (PAPER 1)

__ __ __VU T

S

T

U

P

Q

V


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MATHEMATICS FORM 4

3) Given M and N is the midpoint of the line QR and PS. Namethe angle between the plane VQR and the base PQRS.

SPM 2009 (PAPER 1)

__ __ __NV M

Q

R

U

S

P

V

MN


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SPM 2010 (PAPER 2)

MATHEMATICS FORM 4

Diagram in the answer space shows a right prism. The base CDHG is a horizontalrectangle. TrapeziumABCD is the uniform cross section of the prism.

(a) On diagram in the answer space, markthe angle between the planeBCGF andthe base CDHG .

(b) Hence, calculatethe angle between the planeBCGF and the base CDHG.

[3 marks]Answer :(a)

MarkA

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= 3423


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SPM 2005 (PAPER 2)

MATHEMATICS FORM 4

Diagram shows a right prism. Right-angled triangle PQR is the uniform crosssection of the prism.

(a) Name the angle between the planeRTUandthe planePQTU.

(b) Hence, calculate the angle between

the planeRTU and the basePQTU.[3 marks]

Answer :(a)

(b)

Q R12 cm

18 cm

T

tan RTQ=

RTQ=tan -1

RTQ= 3341

__ __ __TR Q

R

P

Q

S

U

T 12 cm

5 cm

18 cm


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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 pyramidis 11 cm.

Identifyand calculatethe angle betweenthe 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= 6533

Documents

152012746 Form 4 Chapter 11 Lines and Planes in 3 Dimensions (3)