
Vectors in a Plane and Space 



Vectors in
threedimensional space in terms of Cartesian coordinates 
Angles of
vectors in relation to coordinate axes, directional cosines  scalar
components of a vector 
The unit vector of a vector 
Vectors in
a threedimensional coordinate system, examples 





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 represents 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),


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


that are 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. 
Solution:
Let calculate the magnitude or length of the radius vector, 

Angles between the radius vector and the coordinate axes are, 





Example:
A vector AB is directed from point
A(1,
2,
1) to point B(2,
3, 4), find the unit vector of the
vector AB. 
Solution:
Determine the vector AB
from the expression 

The length of the vector
AB 

The unit vector of the vector
AB 




Check
that the directional cosines of the unit vector satisfy the relation, 





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, 





Example: Show 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. 



Example: Points,
A(0,
2,
1), B(2,
1, 3) and
C(3,
1,
2) are the vertices of a triangle, determine the
vector of the median m_{c}
= CM
and its length. 
Solution:
The radius vector of the midpoint of the
side AB, 



The vector of the median
CM,
m_{c}
= CM
= r_{m} 
r_{c}, 

and the length of the median
CM,





To check over the obtained result, calculate the coordinates of the centroid
G, 

The centroid divides every median in the ratio
2 : 1, counting from the vertex to the midpoint, therefore 









Vectors
in 2D and 3D Contents 



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