
Vectors in a Plane and Space 



Vectors
in a Plane 
Unit vector 
Addition, subtraction
and scalar multiplication of vectors, examples 
Linear combination of
vectors 
Linear dependence of
vectors 






Unit vector 
A vector


is called the
unit vector of a vector


if



Therefore,



the unit vector


determines the direction of the vector







Addition subtraction
and scalar multiplication of vectors examples 
Example: 
Given are vectors,


determine






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.

Solution:





Example: Determine the distance of the midpoint
M, of the segment
P_{1}P_{2}, and the point
O, if points,
P_{1}
and P_{2}
are heads of vectors p_{1}
and p_{2} respectively, and whose tails coincide
with the point O
as shows the figure. 
Solution: The
vector P_{1}P_{2}
represents the difference p_{2
}
p_{1}. 






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.



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.



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





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. 
Thus, in the
last 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. 








Vectors
in 2D and 3D Contents 



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