

ALGEBRA


Vectors

::
Vectors
in a plane

Direction,
length or magnitude or norm of a vector

Physical quantities like force, velocity, acceleration
or magnetic field strength, which are defined by a magnitude and a direction, are called vectors or vector
quantities and are represented by directed line segment.



A vector with no
magnitude or, if the tail and the head coincide, is called the
zero or null vector
denoted 



Collinear, opposite
and coplanar vectors

Two vectors are said to be
equal if they have the same magnitude
and direction or if by parallel shift or translation one could be
brought into coincidence with the other, tail to tail and head to
head. 
Vectors are said to be
collinear if they
lie on the same line or on
parallel lines.
Two collinear vectors of the same magnitudes but opposite directions are said to be
opposite vectors. 

Three or more vectors are said to be
coplanar if they lie on the same plane. If two of three vectors
are collinear then these vectors are coplanar. Therefore, if vectors are
parallel to a given plane, then they are coplanar.


Addition of vectors

The sum of vectors, 

can be obtained graphically by placing the tail of the
b
to the tip or head 

of
a
using translation.
Then, draw an arrow from the initial point (tail) of a
to the endpoint
(tip) of b
to obtain the
result. 

The
parallelogram in the above figure shows the addition b +
a
where, to the tip of
b,
by translation, placed is the tail of
a
then, drawn
is the resultant b +
a
by joining the tail of
b
to the tip of
a. 
Note that the tips of the resultant and the second summand should coincide. 
Thus, in the above figure
shown is, the triangle
rule (law) and the parallelogram
rule for finding the
resultant or
the addition of the two given vectors. The result is the same
vector, that is 
a +
b
= b +
a 
vector addition is commutative. 
Since vectors,
a,
b and a +
b
form a triangle, they lie on the same
plane, meaning they are coplanar. 

Addition of three
vectors,
a,
b and c is defined as
a +
b
+
c
= (a +
b)
+
c
and represented graphically

a +
b
+
c
= (a +
b)
+
c 


a +
b
+
c
= a +
(b
+
c) 



The
above diagrams show that vector addition is
associative, that is 
(a +
b)
+
c
= a +
(b
+
c) 

The same way defined is the
sum of four
vectors. 


If by adding vectors obtained is a closed polygon,
then the sum is a null vector. 



By adding a vector
a
to its opposite vector

a,
graphically it leads back to the initial point, therefore 
a +
(
a)
=
a 
a
= 0
thus, the
result is the null vector. 

Subtraction of vectors

Subtraction of two vectors,
a
and b
is defined as addition of
vector a
and 
b,
that
is 
a 
b
= a +
(
b). 
As shows
the right figure, subtraction of two vectors can be accomplished directly. 
By using translation place tails of both vectors at the same
point and connect their
tips. 
Note
that the arrow (tip) of the difference
coincides with the tip of the first
vector (minuend). 




Scalar multiplication
or multiplication of a vector by scalar

Scalar is a quantity which is fully expressed by its magnitude or size
like length, time, mass, etc. as any real number.

By multiplying a vector
a
by a real number
l
obtained is the vector l
a
collinear to
a
but, 
l times longer
then a
if  l
 > 1,
or shorter then a
if  l
 < 1,
and 
directed as a
if l
> 0,
or opposite to a
if l
< 0,
as is shown in the below figure.


Thus, the magnitude of the vector
l
a
equals to the product of the absolute value of the real number
l
and the magnitude of the vector a. 
For the multiplication of a vector by a real number
following rules hold:

1)
l
· ( a + b ) = l
a + l
b 
2)
( l
+ m
) · a = l
a + m
a,
l,
m
Î R 
3)
l
( m
a ) = m
( l
a ) = ( m
l
) a 
4)
1 · a =
a,
1
· a = 
a 
5)
0 · a =
0,
m
· 0 = 0 



In the similar triangles
ABC and
ADE
in the
right 
figure,
AE
: AC
= DE
: BC
= AD
: AB
= l 
therefore,
AE
= l
· AC. 
Since
AE
= l
a + l
b
and AC
= a + b 
then,
l
a + l
b = l
· ( a + b ). 




Unit vector

A vector


is called the
unit vector of a vector


if 


Therefore,



the unit vector


determines the direction of the vector







Example:
Given is a regular hexagon
ABCDEF with the center
O. Express vectors
CD,
BE,
EA,
and

CE,
in terms of vectors, AB
= a
and BC
= b.



Example:
In a triangle
ABC drown are medians as vectors,



Prove
that




Solution: Replacing the sides of the triangle by vectors directed as in
the diagram,






Example:
Use vectors to prove that line segments joining the midpoints of adjacent sides of a quadrangle,
form a parallelogram.

Solution: The sides of the quadrangle ABCD are replaced by vectors, directed as in
the diagram so that the opposite vectors connecting midpoints are, 


As equal vectors are of same lengths and parallel, therefore the line segments connecting the midpoints of any quadrangle, form a parallelogram. 

Example:
Determine a vector which coincides with the angle bisectors of vectors, a
and b
in the diagram.

Solution: The unit vectors of the vectors
a
and b
are,


and they form a rhombus whose diagonal is 


That is, a vector 

will coincide
with 

the angle bisector, while a vector 





defines
the angle bisector of the supplementary angle of the angle between vectors, a
and b.



Linear combination of
vectors

A vector
d
= l
a
+ m
b,
l, m
Î R
denotes the linear combination of the vectors, shown in the
left diagram. 

On the same way, the vector
e
= l
a
+ m
b
+ n
c
represents the linear combination of the vectors, a,
b
and c,
as shows the right diagram in the above figure. 

Example:
Given is a parallelogram ABCD, the midpoint
M
of the side AD
and the intersection point O
of the diagonals. Express vectors,
AO,
BD,
and BM
as the linear combination of vectors, a
= AB
and b
= AD.


Linear dependence of
vectors

Vectors,
a
= OA
and b
= OB
whose points,
O, A and
B
all lie on the same line are said to be linear
dependent, but if the points,
O, A and
B
do not all lie on the same line then a
and b, are not collinear, and are said to be
linear independent.

a)
If a
and b
are linear independent vectors then every vector
d
of the plane determined by a
and b, can be
written as the linear combination of these vectors, that is in the form


Vectors,
a
= OA, b
= OB and
c
= OC,
whose points,
O, A,
B
and C
all lie on the same plane, are said to be coplanar or linear dependent.
But if points, O, A,
B and
C
do not all lie on the same plane, then a,
b
and c
are not coplanar, and are said to be linear independent.


b) If, a,
b
and c are linear independent vectors in 3D space then every vector
d, from the space, can be
represented as


that is, as the linear combination of the vectors,
a,
b
and c. 
In the
below example the sides of the cuboid are replaced by three linear independent vectors,
a,
b
and c
so that the vector MN
can be written as the linear combination of the vectors, a,
b
and c. 

Example:
To a cuboid ABCDEFGH the point
M
divides the edge EH
in the ratio EM
: MH
= 2 : 1 and the point
N
the edge AB
in the ratio AN
: NB
= 3 : 2. Express the vector
MN
as the linear combination of vectors, a
= AB, b
= AD
and c
= AE.

Solution: 

MN
is the linear
combination
of the vectors, a,
b
and c. 

















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