
Vectors in a Plane and Space 

Scalar product or dot
product or inner product 
Orthogonality or
perpendicularity of two vectors 
Different positions of two vectors and the corresponding values of
the scalar product 
Square of magnitude of
a vector 
Scalar product of unit
vectors 
Scalar or dot product
properties 
Scalar product in the
coordinate system 
Angle between vectors
in a coordinate plane 
Scalar product of vectors examples 







Scalar product or dot
product or inner product 
On the beginning of this
section we have already mentioned that there are physical quantities as; force,
velocity, acceleration, electric and magnetic field and so on, which all have vectors’ properties. 
There’s another group of physical quantities as; distance, time, speed, energy, work, mass and so on, which
have magnitude but no direction, called scalar quantities or scalars. 
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. 
The magnitude of any vector is a scalar. 
The
scalar (numeric) product of two vectors geometrically is the product of the length of the first vector
with projection of the second vector onto the first, and vice versa, that is 




The scalar or the dot product of two vectors returns as the result scalar quantity as all three factors on the
right side of the formula are scalars (real numbers). 
The result will be positive or negative depending on
whether is the angle j between the two vectors are acute or obtuse. 
For example, in physics mechanical work
W is the dot product of force
F and displacement
s, 

Obviously, a change of the angle between the two vectors
changes the value of the work W, from the maximal value 
for
j
= 0°
=>
W
=  F  ·

s ,

to the minimal value 
for
j
= 180°
=>
W
=  
F  ·

s . 



For j
= 90° the force
F
does any work on an object, since



It is only the component of
the force along the direction of motion of the object which does any
work. 

Orthogonality or
perpendicularity 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. 


Different positions of two vectors and the corresponding values of
the scalar product 
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 are, 


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, 



 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:
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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