
Vectors in a Plane and Space 

Vectors
and a coordinate system, Cartesian vectors 
Projection of a vector in the direction of another vector,
the scalar and vector components 
The scalar
component 
The vector component 
Scalar product of vectors examples 







Projection of a vector in the direction of another vector,
the scalar and vector components 
The scalar
component 
The length of projection of
a
in the direction of
b
or the scalar component a_{b},
from the diagram, 





Thus, the scalar component of a vector
a
in the direction of a vector b
equals the scalar product of the vector
a
and the unit vector b^{0}
of the vector
b.


The vector component 
By multiplying the scalar component
a_{b}, of a vector
a in the direction of
b, by the unit vector
b^{0
}of the vector
b, obtained
is the
vector component
of a
in the direction of
b. 

or 







Scalar
product of vectors examples 
Example:
Applying the scalar product, prove
Thales’ theorem which states that an angle inscribed in a 
semicircle is a right angle. 
Solution:
According to diagram in
the right figure 


since square of a vector equal to square of its length, 




thus,
a
and b are orthogonal vectors as 

. 


Example:
Prove the law of cosines used in the trigonometry of oblique triangles. 
Solution:
Assuming the directions of vectors as in
the right diagram 

Using the scalar product and substituting square of vectors
by square of their lengths, obtained is 
a^{2} =
b^{2} + c^{2} 
2bc · cosa
 the law of cosines. 



In the same
way, 


Example:
Determine a parameter l so the given vectors,
a
= 2i
+ l
j

4k
and b
= i 
6 j + 3k
to be perpendicular. 
Solution:
Two vectors are perpendicular if their scalar product is zero, therefore 


Example:
Find the scalar product of vectors, a
= 3m
+ n
and b
= 2m 
4n if
 m 
= 3 and
 n 
= 5 , and the
angle between vectors, m
and n is
60°. 


Example:
Given are vertices, A(2,
0, 5), B(3, 3,
2), C(1,
2,
0) and
D(2,
1, 3), of a parallelogram, 
find the
angle subtended by its diagonals as is shown in the diagram below. 
Solution:






Example:
Given are points, A(2,
3,
1), B(3, 1,
4), C(0,
2, 1) and
D(3,
0, 2), determine the scalar and
vector components of the vector AC
onto vector BD. 
Solution:
The scalar component of the vector
AC
onto vector BD, 

The vector component of the vector
AC
in the direction of the vector BD
equals the product of the scalar component AC_{BD}
and the unit vector BD°
that is










Vectors
in 2D and 3D Contents 



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