
Vectors in a Plane and Space 



Vectors in
threedimensional space in terms of Cartesian coordinates 
Vectors in
a threedimensional coordinate system examples 





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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