

ALGEBRA


::
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, 




Vectors in
threedimensional space in terms of Cartesian coordinates

By introducing three mutually perpendicular unit
vectors, i,
j and
k, in direction of coordinate axes of the
threedimensional coordinate system, called
standard basis vectors, every point
P(x,
y, z) of the space
determines the radius vector or the position vector,



Similarly, a vector a
in the right diagram, which is directed from a point P_{1}(x_{1},
y_{1}, z_{1})
to a point
P_{2}(x_{2},
y_{2}, z_{2})
in space, equals to sum of its vector components, a_{x}i,
a_{y }j,
and
a_{z}k,
in the direction of the coordinate axes,
x,
y, and
z
respectively, that is


The vector
a
is the difference of the radius vectors, 


thus 



so the scalar (numeric) components of the vector
a,


Therefore, the length of the vector
a,



Angles of vectors in relation
to coordinate axes, directional cosines  scalar components of a vector

The scalar components of a vector and its magnitude form a right triangle in which the hypotenuse equals the magnitude of the vector, then


since b
= 90° 
a
then cosb
= sina, 
and
components of a vector a, 





If a vector
a
in 3D space forms with the coordinate axes, x,
y and z
angles,
a,
b
and g
respectively, then

components of the vector are,


while for the radius vector of the point
P(x,
y, z),


Thus, for angles, a,
b
and g
hold the expressions,


called the
directional cosines of the vector
a.





The unit vector of the vector
a,



This condition must satisfy angles that a vector in threedimensional space form with the coordinate axes.

For vectors in a coordinate plane


and since
b
= 90° 
a,
then 

cosb
= cos(90°

a)
= sina,
follows 

the basic trigonometric identity.



Similarly, the
unit vector of a radius vector
in
space


determined by corresponding directional
cosines.

Therefore the radius vector forms with coordinate axes
angles,


As vectors are uniquely determined by its components or coordinates, they are usually denoted using
matrix algebra notation, in the coordinate plane, 

and in the 3D
space 



Vectors in
a threedimensional coordinate system, examples 
Example:
Determine angles that a radius vector of the point
A(3,
2, 5) forms with the coordinate axes. 

Example:
A vector
a
in a 3Dspace, of the length  a

= 4, forms with axes,
x and
y
the same angles, a
= b
= 60°, find the
components (coordinates) of the vector a. 
Solution: Using relation 

applying given conditions, 

The components of
a, 


Example: Prove
that vectors, a
= i
+ 3 j
+ k,
b
= 3i 
4 j 
2k and
c
= 5i 
10 j 
4k
are coplanar. 
Solution: If all three vectors lie on the same plane then there are coefficients,
l and
m such that, for example
c
= la
+ mb,
i.e., each of the vectors can be expressed as the linear combination of the remaining two. 

















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