|
|
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
= a2
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 = a2
= | 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 three-dimensional 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 |
| a2 =
b2 + c2 -
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 ACBD
and the unit vector BD°
that is
|
|
|
|
|
|
|
|
|
|
|
|
|
| Vectors
in 2D and 3D Contents |
|
 |
|
| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
