
Vectors in a Plane and Space 



Vectors
and a coordinate system, Cartesian vectors 
Vectors in
a coordinate plane (a twodimensional system of coordinates), Cartesian
vectors 
Radius vector or position vector 
Vector components 
Vectors in
a twodimensional system examples 





Vectors
and a coordinate system, Cartesian
vectors 
Vectors in
a coordinate plane (a twodimensional system of coordinates), Cartesian
vectors 
By introducing a coordinate system in a plane with the unit vectors,
i and
j
(in direction of x
and y
coordinate axis, respectively) whose tails are in the origin O, then each point of the plane determines a vector
r
= OP. 
A directed line segment from the origin to a point
P (x,
y) in plane is called a
radius vector and denoted
r. 
The radius
(or position) vector equals the sum of its
vector components,
xi
and y j
in direction of coordinate axes, that is 


Consider
a vector a
in the plane directed from a point
P_{1}(x_{1},
y_{1})
to P_{2}(x_{2},
y_{2})
shown in the right diagram, it also equals to the sum of corresponding
vector components
a_{x}i
and
a_{y}j,
in direction of coordinate axes,


The diagram shows that 

are the radius vectors of
points P_{1} and
P_{2}, 

thus
the vector a
equals to the difference (joins their heads), that is 

It is obvious from the above diagrams that a vector and its components, i.e., its projections in direction of
coordinate axes, form a right triangle, from which, according to Pythagoras’ theorem, we determine the length of the vector, 


Example:
Determine a vector a whose tail is at the point
P_{1}(4,
1) and head at the point
P_{2}(1,
3).

Solution:
Points, P_{1} and
P_{2} determine radius vectors, 

therefore, 





Example:
At what point P_{1}(x_{1},
y_{1}) has the vector
a
= 7i
+ 2j
its tail, if its head is at the point

P_{2}(3,
4)? 
Solution: 
Using 

or after substitution 



Knowing that two vectors are equal if their corresponding scalar (numeric) components are equal, it follows 


Example:
To a parallelogram given are vertices, A(2,
3), B(4,
2)
and D(3,
5). Determine coordinates of the
vertex C
and the intersection point S
of the diagonals.

Solution:
To the given vertices point radius vectors, 

According to the diagram, the radius vector of the point
C, 





Hence, the intersection point S
of the diagonals S(7/2,
3/2).


Example:
Given are vectors, a
= 2i
+ 3j
and b
= 4i
+ a
j, determine the coefficient
a
such that the vectors to be collinear.

Solution:
In order vectors to be collinear must be 

as two vectors are equal if their corresponding scalar components are equal,
then 




Vectors, a
= 2i
+ 3j
and b
= 4i 
6
j
are collinear, as shows diagram in the above figure.









Vectors
in 2D and 3D Contents 



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