parallelogram principle

parallelogram principle

•

A starting for learning vectors is to think that they are

directed line segments

. Thus a vector

u

→

has a direction and a length (magnitude) and nothing else. Therefore, if two vectors

u

→

and

v

→

have a same direction and a same length, one can consider them identical (and denote

u

→

=

v

→

). So the location of a certain vector in the plane (or in the space) is insignificant; in fact one may also think that this vector consists of all possible directed line segments having a common direction and a common length.

•

However, a vector

u

→

as an

infinite set

of

directed segments

is quite uncomfortable to handle, and one can choose from all possible representants of

u

→

one

individual

directed segment

A

⁢

B

→

, i.e. a

line segment

directed from a certain point

A

(the

initial point

) to another certain point

B

(the

terminal point

). Although

A

⁢

B

→

is only a representant of

u

→

, one may write

A

⁢

B

→

=

u

→

or

u

→

=

A

⁢

B

→

.

•

For describing a vector

u

→

, it’s convenient to know its position in the

coordinate system

of the plane (or the space); there one can say e.g. how great a displacement

u

→

means from left to right (i.e. in the direction of

x

-axis) and how great from below upwards (i.e. in the direction of the

y

-axis); those displacements may be expressed with two numbers.  One may for example write

u

→

=

(

+

5

-

1

)

,

(1)

where the first (upper) number

+

5

tells that the vector leads 5 length-units to the right and the second (lower) number

-

1

that it leads 1 length-unit downwards.

Addition of vectors

Since the vector may be interpreted as a of a horizontal displacement and a vertical displacement, it’s meaningful that by the

addition

of two vectors the horizontal displacements are summed and likewise the vertical displacements.  Accordingly, if we have

v

→

=

(

+

1

-

3

)

,

(2)

then the

sum

of the vectors (1) and (2) is

u

→

+

v

→

=

(

+

5

-

1

)

+

(

+

1

-

3

)

=

(

+

5

+

1

-

1

-

3

)

=

(

+

6

-

4

)

,

which result means a vector leading 6 length-units to the right and 4 down.

When we set the vectors

u

→

and

v

→

one after the other, as in the above picture, and

take the sum vector from the initial point of the first addend to the terminal point of the second addend

, then both the horizontal and the vertical displacements are respectively added.  The addition rule as a

formula

using the points is

P

⁢

Q

→

+

Q

⁢

R

→

=

P

⁢

R

→

.

Note, that

the sum vector

u

→

+

v

→

can be also obtained as the

diagonal vector

of the

parallelogram

with one pair of opposite sides equal to

u

→

and the other pair of opposite sides equal to

v

→

.  The parallelogram picture illustrates also that the vector addition is commutative, i.e. that

u

→

+

v

→

=

v

→

+

u

→

.

If we think the second (dashed in the third picture)

diagonal

of the parallelogram, it is halved by the first (blue) diagonal, since the diagonals of any parallelogram bisect each other (see

parallelogram theorems

); as well the (blue) diagonal representing the sum

u

→

+

v

→

is halved into two equal vectors (better: directed segments)

m

→

=

1

2

⁢

(

u

→

+

v

→

)

.  In the

triangle

A

⁢

B

⁢

C

, the vectors

u

→

,

v

→

,

m

→

may be called two

side vectors

and a

median vector

, all having the common initial point

A

.  Thus we can write the

Theorem

.

In a triangle, the median vector emanating from a certain vertex is the

arithmetic mean

of the side vectors emanating from the same vertex.

Title

parallelogram principle

Canonical name

ParallelogramPrinciple

Date of creation

2013-03-22 17:47:04

Last modified on

2013-03-22 17:47:04

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Topic

Classification

msc 53A45

Synonym

addition of vectors

Synonym

vector addition

Synonym

sum vector

Related topic

CommonPointOfTriangleMedians

Related topic

ProvingThalesTheoremWithVectors

Defines

directed line segment

Defines

sum of vectors

Defines

side vector

Defines

median vector

Defines

diagonal vector

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