aarjaneiro/planetmath huggingface dataset:
Steiner system

Steiner system

Definition

. An

S

⁢

(

τ

,

κ

,

ν

)

Steiner system

is a

τ

-

(

ν

,

κ

,

1

)

design (i.e.

λ

=

1

). The values

τ

,

κ

,

ν

are the parameters of the Steiner system.

Since

λ

=

1

, a Steiner system is a

simple design

, and therefore we may interpret a block to

be

a set of points (

B

=

𝒫

B

), which we will do from now on.

Given parameters

τ

,

κ

,

ν

, there may be several non-isomorphic systems, or no systems at all.

Let

𝒮

be an

S

⁢

(

τ

,

κ

,

ν

)

system with point set

𝒫

and block set

ℬ

, and choose a point

P

∈

𝒫

(often, the system is so

symmetric

that it makes no

difference

which point you choose). The choice uniquely induces an

S

⁢

(

τ

-

1

,

κ

-

1

,

ν

-

1

)

system

𝒮

1

with point set

𝒫

1

=

𝒫

∖

{

P

}

and block set

ℬ

1

consisting of

B

∖

{

P

}

for only those

B

∈

ℬ

that contained

P

. This works because for any

T

1

⊆

𝒫

1

with

|

T

1

|

=

τ

-

1

there was a unique

B

∈

ℬ

that contained

T

=

T

1

∪

{

P

}

.

This recurses down all the way to

τ

=

1

(a

partition

of

ν

-

τ

+

1

into blocks of

κ

-

τ

+

1

) and finally to

τ

=

0

(one arbitrary block of

κ

-

τ

). If any of the

divisibility

conditions (see the entry design (

http://planetmath.org/Design

) for more detail) on the way there do not hold, there cannot exist a Steiner system with the original parameters either.

For instance,

Steiner triple systems

S

⁢

(

2

,

3

,

ν

)

(the first Steiner systems studied, by Kirkman, before Steiner) exist for

ν

=

0

and all

ν

≡

1

or

3

(

mod

6

)

, and no other

ν

.

The reverse construction, turning an

S

⁢

(

τ

,

κ

,

ν

)

into an

S

⁢

(

τ

+

1

,

κ

+

1

,

ν

+

1

)

, need not be unique and may be impossible. Famously an

S

⁢

(

4

,

5

,

11

)

and a

S

⁢

(

5

,

6

,

12

)

have the

Mathieu groups

M

11

and

M

12

as their

automorphism groups

, while

M

22

,

M

23

and

M

24

are those of an

S

⁢

(

3

,

6

,

22

)

,

S

⁢

(

4

,

7

,

23

)

and

S

⁢

(

5

,

8

,

24

)

,
with connexions to the

binary Golay code

and the

Leech lattice

.

Remark

. A

Steiner system

S

⁢

(

t

,

k

,

n

)

can be equivalently characterized as a

k

-

uniform hypergraph

on

n

vertices such that every set of

t

vertices is contained in exactly one edge. Notice that any

S

⁢

(

2

,

k

,

n

)

is just a

k

-uniform

linear space

.

Title

Steiner system

Canonical name

SteinerSystem

Date of creation

2013-03-22 13:05:37

Last modified on

2013-03-22 13:05:37

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 51E10

Classification

msc 05C65

Related topic

Hypergraph

Related topic

IncidenceStructures

Defines

Steiner triple system

Generated on Fri Feb 9 22:00:01 2018 by

LaTeXML
finite plane

finite plane

A

finite plane

(synonym

linear space

(

http://planetmath.org/LinearSpace2

)

) is the finite (discrete) analogue of planes in more familiar

geometries

. It is an

incidence structure

where any two

points

are

incident

with exactly one

line

(the line is said to “pass through” those points, the points “lie on” the line), and any two

lines

are incident with

at most

one

point

— just like in ordinary planes, lines can be

parallel

i.e. not intersect in any point.

A finite plane without

parallel lines

is known as a

projective plane

. Another kind of finite plane is an

affine plane

, which can be obtained from a projective plane by removing one line (and all the points on it).

Example

An example of a projective plane, that of order

2

, known as the

Fano plane

(for projective planes,

order

q

means

q

+

1

points on each line,

q

+

1

lines through each point):

An edge here is represented by a

straight line

, and the inscribed circle is also an edge. In other words, for a

vertex set

{

1

,

2

,

3

,

4

,

5

,

6

,

7

}

, the edges of the Fano plane are

{

1

,

2

,

4

}

,

{

2

,

3

,

5

}

,

{

3

,

4

,

6

}

,

{

4

,

5

,

7

}

,

{

5

,

6

,

1

}

,

{

6

,

7

,

2

}

,

{

7

,

1

,

3

}

Notice that the Fano plane is generated by the triple

{

1

,

2

,

4

}

by repeatedly adding

1

to each entry, modulo

7

. The generating triple has the property that the differences of any two elements, in either order, are all
pairwise different modulo

7

. In general, if we can find a set of

q

+

1

of the integers (mod

q

2

+

q

+

1

) with all pairwise differences distinct,
then this gives a

cyclic representation

of the finite plane of order

q

.

Title

finite plane

Canonical name

FinitePlane

Date of creation

2013-03-22 13:05:34

Last modified on

2013-03-22 13:05:34

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

18

Author

marijke (8873)

Entry type

Definition

Classification

msc 51E20

Classification

msc 05C65

Classification

msc 51E15

Classification

msc 05B25

Related topic

LinearSpace2

Defines

Fano plane

Generated on Fri Feb 9 22:00:45 2018 by

LaTeXML
dodecagon

dodecagon

A

dodecagon

is a

polygon

with twelve sides.

Title

dodecagon

Canonical name

Dodecagon

Date of creation

2013-03-22 14:25:54

Last modified on

2013-03-22 14:25:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:00:57 2018 by

LaTeXML
dodecahedron

dodecahedron

A

dodecahedron

is a

polyhedron

with twelve faces.

Title

dodecahedron

Canonical

name

Dodecahedron

Date of creation

2013-03-22 14:26:10

Last modified on

2013-03-22 14:26:10

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:00:59 2018 by

LaTeXML
enneacontahedron

enneacontahedron

An

\htmladdnormallink

enneacontahedronhttp://aux.planetmath.org/files/objects/5939/

Enneacontahedron

.JPG is a

polyhedron

with ninety faces.

Title

enneacontahedron

Canonical

name

Enneacontahedron

Date of creation

2013-03-22 14:25:49

Last modified on

2013-03-22 14:25:49

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

15

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:01 2018 by

LaTeXML
heptahedron

heptahedron

A

heptahedron

is a

polyhedron

with seven faces.

According to the Steinitz

classification

of f-vectors of 3-polytopes and the Euler polyhedron formula together imply that a convex heptahedron will have one of the followng f-vectors:

•

(

6

,

11

,

7

)

•

(

7

,

12

,

7

)

•

(

8

,

13

,

7

)

•

(

9

,

14

,

7

)

•

(

10

,

15

,

7

)

Title

heptahedron

Canonical

name

Heptahedron

Date of creation

2013-03-22 14:26:24

Last modified on

2013-03-22 14:26:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51E99

Synonym

septahedron

Related topic

Grafix

Generated on Fri Feb 9 22:01:03 2018 by

LaTeXML
hexahedron

hexahedron

A

hexahedron

is a

polyhedron

with six faces. A

regular

hexahedron is a

cube

. Following is a picture of a cube with dashed

line segments

indicating edges (

http://planetmath.org/Edge

) that are not visible.

Title

hexahedron

Canonical

name

Hexahedron

Date of creation

2013-03-22 14:26:26

Last modified on

2013-03-22 14:26:26

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51E99

Classification

msc 51-00

Synonym

cube

Generated on Fri Feb 9 22:01:05 2018 by

LaTeXML
hexecontahedron

hexecontahedron

A

hexecontahedron

is a

polyhedron

with sixty faces.

Title

hexecontahedron

Canonical

name

Hexecontahedron

Date of creation

2013-03-22 14:25:52

Last modified on

2013-03-22 14:25:52

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:07 2018 by

LaTeXML
icosahedron

icosahedron

An

icosahedron

is a

polyhedron

with twenty faces.

Title

icosahedron

Canonical

name

Icosahedron

Date of creation

2013-03-22 14:26:05

Last modified on

2013-03-22 14:26:05

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:08 2018 by

LaTeXML
icosidodecahedron

icosidodecahedron

An

icosidodecahedron

is a

polyhedron

with thirty-two faces.

Title

icosidodecahedron

Canonical

name

Icosidodecahedron

Date of creation

2013-03-22 14:25:57

Last modified on

2013-03-22 14:25:57

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:10 2018 by

LaTeXML
icositetrahedron

icositetrahedron

An

icositetrahedron

is a

polyhedron

with twenty-four faces.

Title

icositetrahedron

Canonical

name

Icositetrahedron

Date of creation

2013-03-22 14:26:02

Last modified on

2013-03-22 14:26:02

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:12 2018 by

LaTeXML
nonahedron

nonahedron

A

nonahedron

is a

polyhedron

with nine faces.

Title

nonahedron

Canonical name

Nonahedron

Date of creation

2013-03-22 14:26:17

Last modified on

2013-03-22 14:26:17

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:14 2018 by

LaTeXML
octahedron

octahedron

An

octahedron

is a

polyhedron

with eight faces.

Title

octahedron

Canonical

name

Octahedron

Date of creation

2013-03-22 14:26:20

Last modified on

2013-03-22 14:26:20

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:16 2018 by

LaTeXML
pentahedron

pentahedron

A

pentahedron

is a

polyhedron

with five faces.

Title

pentahedron

Canonical

name

Pentahedron

Date of creation

2013-03-22 14:26:30

Last modified on

2013-03-22 14:26:30

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:18 2018 by

LaTeXML
properties of regular tetrahedron

properties of regular tetrahedron

A

regular tetrahedron

may be formed such that each of its edges is a

diagonal

of a face of a cube; then the

tetrahedron

has been

inscribed

in the cube.

It’s apparent that a plane

passing through

the

midpoints

of three

parallel edges

of the cube cuts the regular tetrahedron into two

congruent

pentahedrons

and that the

intersection

figure is a square, the midpoint

M

of which is the

centroid

of the tetrahedron.

The angles between the four

half-lines

from the centroid

M

of the regular tetrahedron to the vertices (

http://planetmath.org/

Polyhedron

) are

2

⁢

arctan

⁡

2

(

≈

109

∘

), which is equal the angle between the four covalent bonds of a carbon .  A half of this angle,

α

, can be found from the

right triangle

in the below figure, where the

catheti

are

s

2

and

s

2

.

One can consider the regular tetrahedron as a cone.  Let its edge be

a

and its height

h

.  Because of

symmetry

, a height line intersects the corresponding base triangle in the centroid of this

equilateral triangle

.  Thus we have (see the below ) the rectangular triangle with

hypotenuse

a

, one

cathetus

h

and the other cathetus (

http://planetmath.org/Cathetus

)

2

3

⋅

a

⁢

3

2

=

a

3

(i.e.

2

3

of the median (

http://planetmath.org/Median

)

a

⁢

3

2

of the equilateral triangle — see the

common point of triangle medians

).  The

Pythagorean theorem

then gives

h

=

a

2

-

(

a

3

)

2

=

a

⁢

6

3

.

Consequently, the height of the regular tetrahedron is

a

⁢

6

3

.

Since the area of the base triangle (

http://planetmath.org/EquilateralTriangle

) is

a

2

⁢

3

4

, the volume (one third of the

product

of the base and the height) of the regular tetrahedron is

a

3

⁢

2

12

.

Title

properties of regular tetrahedron

Canonical

name

PropertiesOfRegularTetrahedron

Date of creation

2013-03-22 18:29:39

Last modified on

2013-03-22 18:29:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Topic

Classification

msc 51E99

Synonym

regular tetrahedron

Related topic

Grafix

Related topic

DehnsTheorem

Related topic

Tetrahedron

Generated on Fri Feb 9 22:01:20 2018 by

LaTeXML
tetradecahedron

tetradecahedron

A

tetradecahedron

is a

polyhedron

with fourteen faces

Title

tetradecahedron

Canonical

name

Tetradecahedron

Date of creation

2013-03-22 14:26:08

Last modified on

2013-03-22 14:26:08

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

7

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:22 2018 by

LaTeXML
tetrahedron

tetrahedron

1

Definition

A

tetrahedron

is a

polyhedron

with four faces, which are

triangles

. A tetrahedron is called

non-degenerate

if the four
vertices do not lie in the same plane. For the

remainder

of this
entry, we shall assume that all tetrahedra are non-degenerate.

If all six edges of a tetrahedron are equal, it is called a

regular tetrahedron

. The faces of a regular tetrahedron are

equilateral triangles

.

2

Basic properties

A tetrahedron has four vertices and six edges. These six edges can be
arranged in three pairs such that the edges of a pair do not
intersect. A tetrahedron is always convex.

In many ways, the

geometry

of a tetrahedron is the three-dimensional
analogue of the geometry of the triangle in two

dimensions

. In
particular, the special points associated to a triangle have their
three-dimensional analogues.

Just as a triangle always can be

inscribed

in a unique circle, so too
a tetrahedron can be inscribed in a unique sphere. To find the centre
of this sphere, we may construct the

perpendicular bisectors

of the
edges of the tetrahedron. These six planes will meet in the centre of
the sphere which

passes through

the vertices of the tetrahedron.

The six planes which connect an edge with the

midpoint

of the opposite
edge (see what was said about edges coming in pairs above) meet in the
barycentre (a.k.a.

centroid

, centre of mass, centre of gravity) of the
tetrahedron.

3

Mensuration

Formulas for volumes, areas and lengths associated to a terahedron are
best obtained and expressed using the method of

determinants

. If the
vertices of the tetrahedron are located at the points

(

a

x

,

a

y

,

a

z

)

,

(

b

x

,

b

y

,

b

z

)

,

(

c

x

,

c

y

,

c

z

)

, and

(

d

x

,

d

y

,

d

z

)

,
then the volume of the tetrahedron is given by the following
determinant:

±

1

6

⁢

|

a

x

a

y

a

z

1

b

x

b

y

b

z

1

c

x

c

y

c

z

1

d

x

d

y

d

z

1

|

.

Figure 1:

A regular tetrahedron

Title

tetrahedron

Canonical name

Tetrahedron

Date of creation

2013-03-22 14:26:32

Last modified on

2013-03-22 14:26:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 51E99

Related topic

StateOnTheTetrahedron

Related topic

RegularTetrahedron3

Related topic

Grafix

Related topic

Triangle

Defines

regular tetrahedron

Generated on Fri Feb 9 22:01:24 2018 by

LaTeXML
triacontahedron

triacontahedron

A

triacontahedron

is a

polyhedron

with thirty faces.

Title

triacontahedron

Canonical

name

Triacontahedron

Date of creation

2013-03-22 14:25:59

Last modified on

2013-03-22 14:25:59

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:26 2018 by

LaTeXML
undecahedron

undecahedron

An

undecahedron

is a

polyhedron

with eleven faces.

Title

undecahedron

Canonical

name

Undecahedron

Date of creation

2013-03-22 14:26:14

Last modified on

2013-03-22 14:26:14

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

6

Author

evin290 (5830)

Entry type

Definition

Classification

msc 51E99

Generated on Fri Feb 9 22:01:27 2018 by

LaTeXML
betweenness in rays

betweenness in rays

Let

S

be a

linear ordered geometry

.
Fix a point

p

and consider the pencil

Π

⁢

(

p

)

of all rays
emanating from it. Let

α

≠

β

∈

Π

⁢

(

p

)

. A point

q

is
said to be an

interior point

of

α

and

β

if there
are points

s

∈

α

and

t

∈

β

such that

1.

q

and

s

are on the same side of line

p

⁢

t

↔

, and

2.

q

and

t

are on the same side of line

p

⁢

s

↔

.

A point

q

is said to be

between

α

and

β

if
there are points

s

∈

α

and

t

∈

β

such that

q

is
between

s

and

t

. A point that is between two rays is an
interior point of these rays, but not vice versa in general. A ray

ρ

∈

Π

⁢

(

p

)

is said to be

between

rays

α

and

β

if there is an interior point of

α

and

β

lying
on

ρ

.

Properties

1.

Suppose

α

,

β

,

ρ

∈

Π

⁢

(

p

)

and

ρ

is between

α

and

β

. Then

(a)

any point on

ρ

is an interior point of

α

and

β

;

(b)

any point on the line containing

ρ

that is an interior
point of

α

and

β

must be a point on

ρ

;

(c)

there is a point

q

on

ρ

that is between

α

and

β

. This is known as the

Crossbar Theorem

, the two “crossbars” being

ρ

and a

line segment

joining a point on

α

and a point on

β

;

(d)

if

q

is defined as above, then any point between

p

and

q

is between

α

and

β

.

2.

There are no rays between two opposite rays.

3.

If

ρ

is between

α

and

β

, then

β

is not
between

α

and

ρ

.

4.

If

α

,

β

∈

Π

⁢

(

p

)

has a ray

ρ

between them, then

α

and

β

must lie on the same

half plane

of some line.

5.

The

converse

of the above statement is true too: if

α

,

β

∈

Π

⁢

(

p

)

are distinct rays that are not opposite of
one another, then there exist a ray

ρ

∈

Π

⁢

(

p

)

such that

ρ

is between

α

and

β

.

6.

Given

α

,

β

∈

Π

⁢

(

p

)

with

α

≠

β

and

α

≠

-

β

. We can write

Π

⁢

(

p

)

as a

disjoint union

of
two subsets:

(a)

A

=

{

ρ

∈

Π

⁢

(

p

)

∣

ρ

⁢

is between

⁢

α

⁢

and

⁢

β

}

,

(b)

B

=

Π

⁢

(

p

)

-

A

.

Let

ρ

,

σ

∈

Π

⁢

(

p

)

be two rays distinct from both

α

and

β

. Suppose

x

∈

ρ

and

y

∈

σ

. Then

ρ

,

σ

belong to the same subset if and only if

x

⁢

y

¯

does not

intersect

either

α

or

β

.

References

1

D. Hilbert,

Foundations of Geometry

, Open Court Publishing Co. (1971)

2

K. Borsuk and W. Szmielew,

Foundations of Geometry

, North-Holland Publishing Co. Amsterdam (1960)

3

M. J. Greenberg,

Euclidean

and

Non-Euclidean Geometries

,

Development

and History

, W. H. Freeman and Company, San Francisco (1974)

Title

betweenness in rays

Canonical name

BetweennessInRays

Date of creation

2013-03-22 15:33:05

Last modified on

2013-03-22 15:33:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51F20

Classification

msc 51G05

Related topic

Angle

Related topic

Ray

Related topic

Midpoint4

Defines

interior point

Defines

between rays

Defines

between two rays

Defines

crossbar theorem

Generated on Fri Feb 9 22:01:36 2018 by

LaTeXML
orthogonal

orthogonal

The word

orthogonal

comes from the Greek

orthe

and

gonia

, or “

right angle

.” It was originally used as synonym of

perpendicular

. This is where the use of “orthogonal” in

orthogonal lines

,

orthogonal circles

, and other geometric terms come from.

In the realm of

linear algebra

, two vectors are orthogonal when their

dot product

is zero, which gave rise a

generalization

of two vectors on some

inner product space

(not necessarily dot product) being orthogonal when their

inner product

is zero.

There are also particular definitions on the following entries:

•

orthogonal matrices

•

orthogonal polynomials

•

orthogonal vectors

In a more broad sense, it can be said that two objects are orthogonal if they do not “coincide” in some way.

Title

orthogonal

Canonical name

Orthogonal

Date of creation

2013-03-22 12:07:30

Last modified on

2013-03-22 12:07:30

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

13

Author

akrowne (2)

Entry type

Definition

Classification

msc 51F20

Classification

msc 65F25

Classification

msc 15A63

Classification

msc 05E35

Classification

msc 42C05

Classification

msc 33C45

Classification

msc 15A57

Generated on Fri Feb 9 22:01:41 2018 by

LaTeXML
proving Thales’ theorem with vectors

proving Thales’ theorem with vectors

Let the radius of the circle be

r

and

A

⁢

B

a

diameter

of the circle. We make the

dot product

calculation

P

⁢

A

→

⋅

P

⁢

B

→

=

(

P

⁢

O

→

+

O

⁢

A

→

)

⋅

(

P

⁢

O

→

+

O

⁢

B

→

)

=

(

P

⁢

O

→

+

O

⁢

A

→

)

⋅

(

P

⁢

O

→

-

O

⁢

A

→

)

=

P

⁢

O

→

⋅

P

⁢

O

→

-

O

⁢

A

→

⋅

O

⁢

A

→

=

r

2

-

r

2

=

0

.

The result shows that

P

⁢

A

→

⟂

P

⁢

B

→

, i.e. the

circumferential angle

A

⁢

P

⁢

B

of the half-circle is a

right angle

.

Title

proving Thales’ theorem with vectors

Canonical name

ProvingThalesTheoremWithVectors

Date of creation

2013-03-22 17:47:45

Last modified on

2013-03-22 17:47:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Proof

Classification

msc 51F20

Classification

msc 51F20

Synonym

vector proof of Thales’ theorem

Related topic

ThalesTheorem

Related topic

SumOfVectors

Related topic

DifferenceOfVectors

Related topic

ScalarSquare

Related topic

ParallelogramPrinciple

Generated on Fri Feb 9 22:01:43 2018 by

LaTeXML