LINES AND PLANES IN 3-DIMENSION

To answer the question from this topic, thestudents must acquire the following skills :

• Able to identify the angle between a lineand a plane ( 1 Mark )

• Able to calculate the angle between a line• Able to calculate the angle between a lineand a plane ( 3 Marks ).

• Able to identify the angle between twoplanes ( 1 Mark ).

• Able to calculate the angle between twoplanes ( 3 Marks )

Diagram shows a cuboid witha horizontal rectangular base.Calculate the angle betweenthe planeTWR and the planePSWT.

W

P

V

T

U

8cm

5cm

S

RQ

P1

EXAMPLE :

SOLUTION :

R T / W S

At the back

RQ

Tan RWS =

RWS = Tan -1

= 580

5

8

5

8

K2

N1

W

S R8cm

5cm

E F

GH

ACTIVITY 1 : TO IDENTIFY THE PLANE

AB

CD

PLANE AT THE TOP : PLANE EFGH

CD

E F

GH

AB

CD

PLANE ON THE LEFT : PLANE ADHE

CD

E F

GH

AB

CD

PLANE IN THE FRONT : PLANE ABFE

CD

E F

GH

AB

CD

PLANE AT THE BACK : PLANE DCGH

CD

E F

GH

AB

CD

PLANE AT THE BOTTOM: PLANE ABCD

CD

E F

GH

AB

CD

PLANE ON THE RIGHT : PLANE BCGF

ON TOP OF THE RED DOT

TO THE RIGHT OFTHE RED DOT

THE LOCATION OF THE POINT

THE RED DOT

IN FRONT OF THERED DOT

AT THE BACK OFTHE RED DOT

ON THETOP OF ….

AT THEBACK OF ….TO THE

LEFT OF ….

AT THEBOTTOM OF ….

IN FRONTOF ….

TO THERIGHT OF ….

ACTIVITY 2 : TO DETERMINE THE LOCATION OF A POINT

A B

CD

EF

GH

POINT TO THE LEFT OF F : POINT EPOINT TO THE LEFT OF F : POINT E

POINT G

POINT AT THE BOTTOM OF F :

POINT AT THE BACK OF F :

POINT TO THE RIGHT OF D :

POINT ON TOP OF D :

POINT IN FRONT OF D :

POINT B

POINT C

POINT H

POINT A

ANGLE BETWEENA LINE AND A PLANE

ALINE

BC

LINE

PLANE

CD

E F

Normal

H G

Activity 3 :To Identify The Angle Between Line And Plane

The line draw fromG andperpendicular tothe plane ABCD iscall normal

BA

Orthogonalprojection

The line lies on theplane ABCD whichjoint the point A to theline GC is known asthe orthogonalprojection of line AGon the plane ABCD.

The angle between the line AGand the orthogonal projection, ACis the angle between the line AGand the plane ABCD that is

GAC.

CD

E F

H G

ACTIVITY 3 : To Identify The Angle Between A Line And A Plane

AG C

At the bottom

Normal

Example 1a

BA

Angle between the line AG and the plane ABCD

Name the angle between the lineAG and the plane ABCD

Orthogonalprojection

= GAC.

EXAMPLE 1(b)

CDE

H G

F

BA

CD

Diagram 1(b)

Diagram 1b shows a cuboid ABCDEFGH.Name the angle between the line HB and theplane ABCD.

ACTIVITY 4 :

To find the angle between a line and a plane

Example 2(a)

5cm

CD

E

H G

F

12cm

5cm

Diagram 2(a) shows a cuboid, ABCDEFG. Find theangle between the line AH and the plane DCGH.

BA

CD

4cm

Diagram 2a

Draw the line AH andshade the plan DCGHin diagram 2a.

1.

SolutionsStepsNo

CD

E

H G

F

5cm

4cm

12cm

A BDiagram 2a

Diagram 2a shows a cuboid, ABCDEFG. Find the angle between the lineAH and the plane DCGH.

No Steeps Solutions

2 Use the method youhave learned in activity3, identify the anglebetween the line AHand the plane DCGH

HA

back

D

5cm

BA

CDE

H G

F

4cm

12cm

No Steps Solutions

3 Refer to the points you haveobtained in steep 2 (point A, H,D), complete the ∆ AHD. Mark

AHD. Mark the right angle,

HDA. Transfer out the

∆ AHD.

H

DA

∆ AHD.

5cm

4cm

A H D

BA

CDE

H G

F

12cm

No Steps Solutions

4 With the information given in thequestion, label the length of thesides of ∆ AHD. At least the length for 2 sides must be known.Use Pythegoras Theorem ifnecessary.

5cm

BA

CDE

H G

F

4cm

12cm

No Steps Solutions

6 Mark,- the opposite side, AD asT- the adjacent side, HD as S

H

DA

5 cm

4 cm

S

TT

5cm

BA

CDE

H G

F

4cm

12cm

No Steps Solutions

6 Use the tangent formula to

calculate AHD.

Remember, use

-The sine formula, if O and H were

known

- The cosine formula, if A and H

Tan AHD =

AHD = tan -1

AHD = 38040’

- SOH

H

OS

5

4

5

4

- The cosine formula, if A and H

were known

-The tangent formula, if O and A

were known

– TOAA

OT

– CAHH

AC

5cm

BA

CD

E

H G

F

4cm

12cm

example 2 (b)

E F

GH

4 cm

12 cm

Diagram 2b shows a cuboid,ABCDEFGH. Calculatethe angle between the line HB and the plane BCGF

A B

CD

3 cm

Diagram 2b

ANGLE BETWEEN TWOPLANESPLANES

ACTIVITY 5 : To Identified The Angle Between TwoPlanes

E

CD

H G

F

EXAMPLE 3(a)

A

B

Diagram 3a

Diagram 3a shows a cuboid,ABCDEFGH. Name the angle betweenthe plane AGH and the plane ABCD

1.DRAW 3 LINES

CDE

H G

F

Bottom

ACTIVITY 5 : To Identified The Angle Between Two Planes

BA

Diagram 3a

Bottom

2.Mark the location(direction) of theplane ABCD at thebottom of the firstline to the left.

Diagram 3a shows a cuboid,ABCDEFGH. Name the anglebetween the plane, AGH and theplane, ABCD

CDE

H G

F

A

Bottom

ACTIVITY 5 : To Identified The Angle Between Two Planes

BA

Diagram 3a

3.Refer to the plane,AGH, identify thepoints whichtouch the plane,ABCD and write itat the middle line.

Diagram 3a shows a cuboid,ABCDEFGH. Name the anglebetween the plane, AGH and theplane, ABCD

CDE

H G

F

AH / G

Bottom

ACTIVITY 5 : To Identified The Angle Between Two Planes

BA

Diagram 3a

3.Refer to the plane,AGH, identify thepoint which doesnot touch theplane, ABCD andwrite it at the firstline to the left.

Diagram 3a shows a cuboid,ABCDEFGH. Name the anglebetween the plane, AGH and theplane, ABCD

CDE

H G

F

AH/G

Bottom

5.Between the point H

ACTIVITY 5 : To Identified The Angle Between Two Planes

BA

Diagram 3a

5.Between the point Hand G, point which isnearer to point A orlocated on the sameplane as point A willbe choosen. Pointwhich is not choosenwill be earased.

Diagram 3a shows a cuboid,ABCDEFGH. Name the anglebetween the plane, AGH and theplane, ABCD

CDE

H G

F

AH

Ke Bawah

5.Between the point H

ACTIVITY 5 : To Identified The Angle Between Two Planes

BA

Diagram 3a

5.Between the point Hand G, point which isnearer to point A orlocated on the sameplane as point A willbe choosen. Pointwhich is not choosenwill be earased.

Diagram 3a shows a cuboid,ABCDEFGH. Name the anglebetween the plane, AGH and theplane, ABCD

CDE

H G

F

AH

Bottom

D

6. Identify the pointwhich is located at

ACTIVITY 5 : To Identified The Angle Between Two Planes

BA

Diagram 3a

which is located atthe bottom of thepoint H ( )andwrite it on the firstline to the right.Diagram 3a shows a cuboid,

ABCDEFGH. Name the anglebetween the plane, AGH and theplane, ABCD

CDE

HG

F

Bottom

AH D

ACTIVITY 5 : To Identified The Angle Between Two Planes

BA

Diagram 3a

7. In the diagram 3a,complete the ∆ HAD and mark the HAD

Angle between the plane, AGH and the plane, ABCD= HAD

EXAMPLE 3(b)

E5cm

CD

H G

F

12cm

Diagram 3bBA

CDF

4cm

Diagram 3b shows a cuboid with horizontalrectangle base ABCD. Name the anglebetween the plane ACH and the plane CDHG

THANK YOUTHANK YOU