Some technical terms to know aboutSome technical terms to know about

A. Angle between two lines.

B. Angle between a line and a plane.

C. Intersection of two planes.

D. Angle between two planes.

E. A line of greatest slope.

A. Angle between two lines.

B. Angle between a line and a plane.

C. Intersection of two planes.

D. Angle between two planes.

E. A line of greatest slope.

A. Angle between two lines

Whenever ambiguous, always refer to the acute angle.

• A’ is the projection of the point A onto the plane if AA’ is perpendicular to

A

A’

• Line segment A’B’is the projection of A on the plane

A

B

A’ B’

• If the line AB intersects the line at A, then we say that B’ is the projection of B on and AB is the line of projection

A B’

B

(a) If the line is parallel to the plane, we cannot find the point of intersection

L

(b) A normal of a plane is a line perpendicular to every line lying on the plane

L

L2

L3

L1

(c) If a line is not parallel to the plane, then we can find the point of intersection.

L

B. Angle between a line and a plane

A

B C

D

P

Q

N

QN is a perpendicular to the plane ABCD.

PN is the projection of

PQ on the plane ABCD

is the angle between

the line PQ and the

plane ABCD.

(a)Two parallel planes

1

2

Eg. Two walls which are

opposite to each other

(b) Line of intersection of the planes

1 E.g.

2

C. Angle between two planesFind the line of intersection.

Construct two lines which lie on the planes such that both of them are perpendicular to the line of intersectionCan you now define the angle between two planes?

• If a horizontal plane which intersects an inclined plane ( ) at AB and L is a line on , L is the line of greatest slope on if L is perpendicular to AB.

L

A B

• In the figure , XYAB and XYCD are two intersecting planes which XYCD is inclined and XYAB is horizontal , PQ is a line on XYCD and perpendicular to XY。

• Notice that > • The angle between the line of greatest slope

and the horizontal plane is also the angle between the planes

A

D C

X Y

B

P

Q

R

C

O

A B

D

N

 

Find the angle between OB and ABCD.

What is the projection of OB on the plane ABCD?

What is the angle between OB and ABCD?

O

A

D

N M

B

C

Find the angle between ON and plane OBC

What is the projection of ON and plane OBC?

What is the required angle?

A B

D

N

O

C

Find the angle between planes OBC and ABCD

What is the line of intersection?

Draw a line on the plane OBC and BC⊥

M

Draw a line on the plane ABCD and

BC.⊥

What is the required angle?

O

A

D

N M

B

C

Find the angle between planes ONB and ONM

What is the line of intersection?

Draw a line on ONB and ON⊥

Draw a line on ONM and ON⊥

What is the required angle?

O

A

D

N M

B

C

Angle between planes OAB and OBC

What is the line of intersection?

Draw a line on OAB and OB⊥

Draw a line on OBC and OB⊥

What is the required angle?

The EndThe End