

ALGEBRA


::
Scalar product or dot
product or inner product

The word scalar derives from the English
word "scale" for a system of ordered marks at fixed intervals used in measurement, which in turn is derived
from Latin scalae  stairs. 
Physical quantities like, distance, time, speed, energy, work, mass and so on, which have magnitude but no direction, are called scalar quantities or scalars.
Thus, the magnitude of any vector is a scalar. 


Perpendicularity (or
orthogonality) of two vectors

Therefore, if the scalar product of two vectors,
a
and b
is zero, i.e., 
a ·
b
= 0
 then the two vectors are orthogonal. 
And inversely, if two vectors are perpendicular, their scalar product is zero. 


The value of the scalar product of two vectors depends
on their position

Different positions of two vectors and the corresponding values of their scalar product are shown in
the below figures. 






Square of magnitude of
a vector

The scalar product of a vector with itself,
a ·
a
= a^{2}
is the square of magnitude of a vector, that is 

thus, 


in the coordinate plane, and 


in 3D space. 

Scalar product of unit
vectors

The unit vectors,
i,
j and k,
along the Cartesian coordinate axes are orthogonal and their scalar products 

and 



Scalar or dot product
properties

a)
k
· (a · b)
= (k · a) · b
= a · (k · b),
k
Î
R 
b)
a
· b = b · a 
c)
a
· (b + c)
= a · b + a
· c 
d)
a
· a = a^{2}
=  a ^{2} 



According to the definition of the dot product, from the
above diagram, 

then 


what confirms the distribution law. 

The associative law does not hold for the dot product of more vectors, for example 
a
· (b · c) is not equal
(a
· b) · c 
since
a
· (b · c) is the vector
a
multiplied by the scalar b
· c,
while (a
· b) · c
is the vector c
multiplied by the scalar a
· b. 

Scalar product in the
coordinate system

The scalar product can be expressed in terms of the
components of the vectors, 

thus, 

 the scalar product in threedimensional coordinate system, 



 the scalar product in the coordinate plane. 


Angle between vectors
in a coordinate plane

From the scalar product
a
· b = 
a 
·  b  · cosj
derived are, the angle between two vectors, formulas 

for plane vectors 

for space vectors 



The cosine of the angle between two vectors,
a
and b
represents the
scalar product of their unit
vectors,




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: The right figure
shows that 


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




therefore
a
and b are
perpendicular vectors as
a
· b = 0. 

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
such that the 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 


::
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.

















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