Perpendicularity

Perpendicularity

If a line is perpendicular to a plane, it is perpendicular to every line in the plane.

Projection of right angle’s arms are perpendicular lines if one arm of right angle is
parallel to projection plane and the second arm is not perpendicular to it.

D:

p

, m II

p

m' II m

f

m

f

'



m'

fp

n

1,2,3...

f

m



n

1,2,3...

m'



n'

1,2,3...

˜

˜

m

m'

p

f

f

‘=n

1,2,..

'

n

1

n

2

n

3

˜

m

m'

p

f

˜

Monge’s projection

Perpendicular elements

1. Perpendicular lines a



b

D: a II

p

1

a'

a"

X

12

b'

a'



b'

D: a II

p

1

a'



b'

a



b=1

a'

a"

b'

b"

1'

1"

X

12

1. Perpendicular lines a



b

D: a II

p

2

a'

a"

X

12

1. Perpendicular lines a



b

b"

a"



b"

D: a II

p

2

a"



b"

a



b=1

a'

a"

b'

1'

1"

b"

X

12

1. Perpendicular lines a



b

2. Line perpendicular to plane a



a

m

n

a

a

a

1

a

2

p

˜

˜

˜

1

a



m

a



n

m

 a

n

 a

m



n=1

a

a



a



m,n,p...

m,n,p...

 a

A line is perpendicular to a plan if the line is perpendicular to two nonparallel lines
from this plane.

D:

ap

1

a

'

X

12

2. Line perpendicular to plane a



a

a'

a'

a

'

D:

ap

1

a'

a

'

a II

p

1

a

'

a'

a"

X

12

2. Line perpendicular to plane a



a

D:

a

=ABC,

ap

1

a

'

A'

B'

C'

A"

B"

C"

X

12

2. Line perpendicular to plane a



a

– an example

D:

a

=ABC,

ap

1

a'

a

'

a

'

A'

a'

B'

C'

A"

B"

C"

X

12

2. Line perpendicular to plane a



a

– an example

D:

a

=ABC,

ap

1

a'

a

'

a II

p

1

a



ABC

a

'

A'

a"

a'

B'

C'

A"

B"

C"

X

12

2. Line perpendicular to plane a



a

– an example

D:

a

=ABC,

ap

1

a'

a

'

a II

p

1

a



ABC

a

a

=1

a

'

A'

a"

a'

B'

1'

C'

A"

B"

C"

1"

a

'

A'

a"

a'

B'

C'

A"

B"

C"

X

12

2. Line perpendicular to plane a



a

– an example

D:

a

=ABC,

ap

1

a'

a

'

a II

p

1

a



ABC

a

a

=1

a

'

A'

a"

a'

B'

1'

C'

A"

B"

C"

1"

X

12

2. Line perpendicular to plane a



a

– an example

D:

ap

2

a

"

2. Line perpendicular to plane a



a

X

12

a"

a

"

a"

a II

p

2

a'

D:

ap

2

a"

a

"

a II

p

2

a

"

a"

a'

2. Line perpendicular to plane a



a

X

12

D:

a

=ABCD,

ap

2

a

"

A"

B"

C"

D"

2. Line perpendicular to plane a



a

– an example

X

12

A'

B'

C'

D'

a"

a

"

a"

a II

p

2

a



ABCD

a'

D:

a

=ABCD,

ap

2

a"

a

"

a II

p

2

a



ABCD

a

a

=1

a

"

A'

a"

a'

B'

1'

C'

A"

B"

C"

1"

D'

D"

a

"

A'

a"

B'

C'

A"

B"

C"

D'

D"

2. Line perpendicular to plane a



a

– an example

X

12

D:

a

=ABCD,

ap

2

a"

a

"

a II

p

2

a



ABCD

a

a

=1

a

"

A'

a"

a'

B'

1'

C'

A"

B"

C"

1"

D'

D"

X

12

2. Line perpendicular to plane a



a

– an example

3. Plane perpendicular to plane

a



b

D:

a

a

1,2..

 a

a

1,2..

 b

1,2,3...



b

1,2,3...

a

˜

M

a

b

1

b

2

b

3

˜

m

1

m

2

a

1

a

2

˜