1. Introduction
The dot product (also scalar product) is one of the most important multiplications of vectors. It is not only widely used in analytic geometry, vector analysis, linear algebra, mechanics [1] [2] [3] . but also widely used in other fields such as engineering, computer graphs [4] [5] [6] . But there is something imperfect, since dot product does not have corresponding division. Almost everyone spends some time to consider this problem when starting to learn dot product. Then obtains a sad result: the division on dot product does not exist. As a result, there are no papers which successfully present the quotient of a number and a vector as the inverse operation of a dot product.
Our purpose of this paper is to set up a theory to solve the problem that dot product has no corresponding division in 3-dimensional space. Dot product is quite different to cross product. The former exits in each vector space, but the latter just exists in some special vector space [7] [8] . Fortunately, they all exist in three dimensional space. As most of people think the divisions on cross products do not exist, in 2022, Mr. Wang and Ms. Chen successfully built the theory of indefinite cross divisions in 3-dimensional space to have solved the problem that cross product has no corresponding division, by adding angle as parameter [9] . Similar to cross products, we are sure that we can solve the problem that dot product has no corresponding division by adding parameters. We naturally think again that, when computing dot product, the something important we unconsciously ignore includes angles, since there are many books in which the definition of dot product of two vectors
and
is directly defined by their coordinates such as
(1)
where
and
[4] [10] . Though the definition is simple and useful when computing dot products, it somehow hinders us to find the truth of inverse operation of dot product. If we want to inversely find the exact vector
from scalar c and vector
such that
, we will face the fact that there are many many a’s such that
, which may be the main reason that makes us obtain that dot product does not have corresponding division. Fortunately some books stress the angle by presenting the following definition:
Let
and
be two vectors, and
be the angle between
and
. The dot product (also called scalar product) of two vectors
and
, denoted by
, is a scalar, defined as
(2)
where
and
indicate the magnitudes of
and
respectively.
The above definition can be easily found in the internet and in the most books related to vector analysis [1] [2] [3] . It is seen that if the angle is 0 or π then we
can inversely get the exact
by
such that
where
. But
for other angles, we can not. We always think the dot product should have corresponding division like cross product. Thus, a problem naturally arises: Is there another hidden factor that we unconsciously ignore? We notice the fact: If
and
are known, not only the angle between
and
is specified, but also the cross product
is specified in three dimensional space. We find that, the direction of
is the second thing which is unconsciously ignored. However, when we inversely want to obtain
from c and
such that
, we do not know the angle and the direction. If we grasp them and put them as parameters, we then establish the theory of indefinite dot quotients in three dimensional space. In fact, it is enough for us to inversely obtain the exact
from a constant c and a nonzero vector
such that
when we
know an angle
and a direction
.
We claim that there exist inverse operations on dot products if enough conditions are known. If we put unknown enough conditions as parameters we obtain indefinite dot quotients which can solve the problem that dot product has no corresponding division in three dimensional space.
In order to realize our purpose, this paper is arranged as 4 main sections. In Section 2, the definitions of indefinite dot quotients are introduced, and some basic properties are presented. In Section 3, some useful operations of indefinite dot quotients are discussed. In Section 4, the structures of indefinite dot quotients are studied in different angles. In Section 5, some useful coordinate formulas of indefinite dot quotients are simply obtained by their structures. And two simple examples are presented not only to support coordinate formulas but also to show that if we know sufficient information, we really can back to find the exact
such that
from c and
.
2. Indefinite Dot Quotients
In this section, we build the foundation of this paper: indefinite dot quotients and their basic properties. Before introduction of them, we stress that:
on computation of dot product when
since
if and
only if
or
or
.
Definition 2.1. Let c be a scalar and
be a nonzero vector, and let
be an angle parameter such that
, and
be a normal vector parameter such that
. The vector, denoted by
(
), is called left (right) indefinite dot quotient (or division) of scalar c and vector
, if its magnitude is defined as
and its direction, when
, is determined by the following 3 steps:
Step 1. In Figure 1, let O be any point in 3D space, and set
,
.
Step 2. Spread the left (right) hand, satisfying that five fingers are on the plane BON and the thumb is perpendicular to other 4 fingers; and pointing the thumb direction along
and other four fingers along
.
Step 3. The left (right) open hand rotates around vector
through the angle
. Then, the direction that the four fingers point out is just the direction
of
(
).
The left and right indefinite dot quotients are collectively called the indefinite dot quotients (or divisions), simply dot quotients (or divisions).
It is seen that
when
and
. Notice that the above definition does not include the case of
, because it makes the denominator of
to be 0 so that the problem becomes complicated. We will present the supplementary definition on this special case in Section 4. Thus, in the rest of this paper, if no special statement, as we meet the notation
or
, we always suppose
,
,
,
and
.
From Definition 2.1, when
, we have
(3)
Moreover, we have the following simple attributes:
(2.1)
.
(2.2) There is a real number
such that
,
.
(2.1.1)
,
.
(2.1.2)
.
(2.1.3) The ordered three vectors
,
,
obey the right hand rule, and
,
,
also obey the right hand rule.
(2.3) If two vectors
and
have the same direction, then
,
.
(2.4)
,
(Conversion Formulas).
(2.5)
,
(Inverse Formulas).
(2.6)
,
(Angle Formulas).
The attributes of (2.4), (2.5) and (2.6) can be easily understood by Figure 2. There is a point, we should note, not to use
, since
in (2.6).
Note that, the definition of left indefinite dot quotient ensures that for any
,
is a vector such that
. Conversely, we have
Theorem 2.1. Let
be a vector and
be a scalar. If there is a vector
such that
and
, then there is the unique angle
such that
where
is any nonzero vector which has the same direction of
.
Proof. Since
, the angle
between
and
is in
. And since
,
is neither 0 nor π. Let
. According to the definition of left indefinite dot quotient, we have the following three items:
1)
,
,
.
2) The three ordered vectors
,
,
obey the right hand rule; and
,
,
also obey the right hand rule.
3)
and
.
The above three items imply that
and
have the same direction, and
is unique.
Since
, they have the same magnitude
. Thus
.
Figure 2. Conversion of left and right dot quotients.
Theorem 2.2. Let
be a vector and
be a scalar. If there is a vector
such that
and
, then there is the unique angle
such that
where
is any nonzero vector perpendicular to
.
Proof. Let
. Since
,
is one of 0 and π. If
, then
, that suggests
and
have the same direction. If
, then
, that suggests
and
have the opposite direction. For any nonzero vector
, according to the definition of left indefinite dot quotient,
and
are colinear; furthermore, when
,
and
have the same direction, and when
,
and
have the opposite direction. Thus,
and
have the same direction, and
is unique. Since
,
and
have the same magnitude
. Thus
. □
Corollary 2.1. Let
be a vector and
be a scalar. If there is a vector
such that
, then there are an angle
and a nonzero vector
such that
.
Proof. It can be seen from Theorem 2.1 and Theorem 2.2. □
Symmetrically, for right indefinite dot quotients, we have the following three results:
Theorem 2.3. Let
be a vector and
be a scalar. If there is a vector
such that
and
, then there is the unique angle
such that
where
is any nonzero vector which has the same direction of
.
Theorem 2.4. Let
be a vector and
be a scalar. If there is a vector
such that
and
, then there is the unique angle
such that
where
is any nonzero vector perpendicular to
.
Corollary 2.2. Let
be a vector and
be a scalar. If there is a vector
such that
, then there are an angle
and a nonzero vector
such that
.
3. Operations
Since a dot quotient involves four factors: a scalar and two vectors and an angle parameter, the rules of multiplications between scalars and dot quotients become very complicated. Thus, in this section, it is fully necessary to further study them. For the symmetry of left and right dot quotients, we only prove the properties with respect to left dot quotients. We always suppose that all factors involving dot quotients make the expressions valid.
For a real number
, in algebra, we have
,
,
etc. Can these properties be extended to dot quotients? In this section, if no special statement, we always assume
. We have the following properties:
Theorem 3.1. For any
,
,
.
Proof. When
, since
and
have the same magnitude and direction,
. When
, based on the previous result and attribute (2.5),
. □
Theorem 3.2. 1) For any
,
,
.
2) For any
,
,
.
Proof. (The proof of left equation of (1))
For any
, the magnitudes of
and
are equal, because
.
Moreover, when
, since
,
and
have the same direction by the definition of left indefinite dot quotient. Thus
.
(The proof of left equation of (2)) For any
, we have
.
By the attribute (2.5),
;
by the attribute (2.6),
. □
Corollary 3.1.
1) For any
,
,
.
2) For any
,
,
.
Proof. Obvious from Theorem 3.1 and Theorem 3.2. □
Corollary 3.2.
1) For any
,
,
.
2) For any
,
,
.
Proof. 1) If
, then
.
2) If
, then
. □
Theorem 3.3. If
for
, then
1)
; 2)
.
Proof. (1) Since
for
, we have
. Hence there is a real number
, satisfying
. Thus,
Similar to (2). □
Note that, if
, then the above results are not valid, since
and
can not hold at the same time.
4. Structures of Dot Quotients
In this section, we discuss the structures of dot quotients in two kinds of normal vector parameters: one is fixed, and another is not. We always assume that,
is given, and c is a scalar, and
always indicates a normal vector parameter such that
. Then we have the following geometric properties:
Theorem 4.1. Let
and
be fixed. In Figure 3, let O be a point in 3-dimensional Space, and make
and
. Take
and
. For
, through
, draw a straight line
parallel to vector
. Then
1) Point
is on the straight line
if and only if there exists a
such that
or
;
2) Point
is on the straight line
if and only if there exists a
such that
or
.
Proof. (Proof of (1)) In fact, when
is on the right of line
in Figure 3,
, where
. According to Corollary 2.1, there exists a
such that
. Similarly, when
is on the left of line
(that is,
), there exists a
such that
.
Conversely, if there exists a
such that
or
. This tells that
. And since
, we have
. Moreover
. This implies that
. Thus,
is parallel to
. Therefore
is on the line
.
Similarly, we can prove (2). □
Corollary 4.1. Let
and
be fixed. In Figure 3, the point sets
and
form two parallel lines, whose distance is
.
Proof. Obvious. □
Corollary 4.2. Let
and
be fixed. In Figure 3, the point sets
and
form two parallel rays on the right parts of
and
, whose distance is
. Symmetrically, the point sets
and
form two parallel rays on the left parts of
and
, whose distance is also
.
Proof. Obvious. □
Corollary 4.3. Let
and
be fixed. Given
, for any
such that
and
, there is a real number
such that
and
.
Proof. They can be found from Figure 3. □
It is readily seen that, if
, then
is required; and if
, then
.
Theorem 4.2. Let
and
be fixed. Then, for any
such that
, there is a real number
satisfying
(4)
where
.
Proof. When
, we have
. According to Figure 3, we can take
. Then there is a real number
such that
. Hence, on the one hand,
. On the other hand,
. Thus,
. Since
, by Equation (3), we have
with
.
When
, we have
. In the previous proof, substituting
for
and
for
, we also have
and
.
Symmetrically, we have
. □
Corollary 4.4. Let
and
be fixed. For any
with
, there is a real number
such that
.
Proof. Obvious. □
Corollary 4.5. Let
and
be fixed. For any
with
, there is a real number
such that
.
Proof. Obvious. □
Because Definition 2.1 does not include the case of
, it leads to something imperfect. How to define dot quotients on this special case?
In Figure 3, let
and
and
, and let
and
,
. We can find that, if we do not change the directions and magnitudes of four vectors
,
,
and
, when c goes to 0, not only the vectors
and
(
and
) are all closing to the vector
(
), but also
and
are all closing to
. In other words, we have the following two facts:
1)
, and
.
2)
and
.
Thus, according to the above discussion, we can present a supplementary definition of indefinite dot quotients for the case of
to complete our theory.
Definition 4.1. Let
be an arbitrary nonnegative scalar.
is defined as
,
that is,
and
is defined as
, that is,
where
is called a scalar parameter (see Figure 4).
For
and
, Attribute (2.1)-(2.6) hold. In fact, they can be observed by Figure 4. For instance, Attribute (2.6) becomes
,
,
that is,
and
. However, the most of results, involving the multiplications of scalars and numerators, in Section 3 do not hold. In other words, the properties except Theorem 3.1 do not hold. Since, at this time, the new numerator as the multiplication of a scalar and an old numerator is 0 which leads to invalid properties. For instance, the result: for
,
,does not hold. In fact, let
. Then
.
If we know more valid information about
and
, then we can imply the exact them. We have
Theorem 4.3. If there is a real number
such that
or
, then
Proof. (using condition
) According to the above definition, there is a real number
such that
. We then have,
.
Similarly, we can get the other equation.
If we use condition
, we can obtain the same results. In fact, condition
is equivalent to condition
. □
Corollary 4.6. For any
, there is a real number
such that
and
(5)
where
if there is a real number
such that
or
when
.
Proof. By Theorem 4.2 and Theorem 4.3. □
Theorem 4.4. Let
such that
. If
(
), then there is a
(
) satisfying
and
where
and
and
and
.
Proof. (We only prove the case of
) If one of
and
and c is 0, or
, the results hold obviously.
Let us simply assume
. And let
. Since
,
. In Figure 5, let O be a point in 3-dimensional Space, and make
and
. Take
and
. And take
and
. And more, take
and
. From Theorem 4.1, the three points
and
and
are on the same line, which is the exact line
in Figure 3. Since
, we have
, that leads to
. Since
,
, which shows
.
Thus, E is on the line
. From Theorem 4.1, there is a
satisfying
, that is,
.
By symmetry, we have
. □
In the above theorem, interval
can not be extended to
. In fact, if
, we have
. Let
, and let
with
. Then, there is no
satisfying
, since
and
for any
. Interval
, of course, can not be extended to
.
Corollary 4.7. Let
be s nonnegative real numbers such that
. For
, if
(
), then there is a
(
) satisfying
and
where
and
and
.
Proof. It can be proved by mathematical induction based on Theorem 4.4.
□
In the application, in order to fit new situation, we need to adjust direction parameter to find good new indefinite dot quotients. Considering that the direction parameter
is just perpendicular to
, we have the following geometric properties:
Theorem 4.5. Let
. In Figure 6, let O be a point in 3-dimensional Space, and
. Take
, and
. For
, through
, draw a plane
perpendicular to vector
. Then
1) point
is on the plane
if and only if
and
;
2) point
is on the plane
if and only if
and
.
Proof. 1) Let
be an arbitrary point on the plane
. If
is
, the result, of course, holds. If
is not
, then
. Thus
. At this case,
is obvious.
Conversely, if
and
, Then,
. This means
is on the plane
.
Similarly, 2) can be proved. □
Corollary 4.8. In Figure 6, 1) point
is on the plane
if and only if there are a
and an
such that
or
;
2) point
is on the plane
if and only if there are a
and an
such that
or
.
Proof. Easy. □
In Figure 6, let
be a fixed angle. Then the point sets
and
form two cycles. One’s center is
, and another’s is
.
In Figure 6, we can find that, when
, two planes
and
are all closed to the same plane through O and
. Thus, we have
Theorem 4.6. Let
. In Figure 7, let O be a point in 3-dimensional Space, and
. Through O, draw a plane
perpendicular to vector
. Then point P is on the plane
if and only if
.
Proof. Obvious. □
In Figure 7, given
, the point set
forms a cycle in the plane
, whose center is O and radius is r.
5. Coordinates of Dot Quotients
Since our theory on dot quotients is built earlier in 3-dimensional Space, which does not use any coordinate system, it holds widely. For simplicity, in this section, we just consider the coordinate formulas of dot quotients in some given rectangular coordinate system
. Then, from the coordinates of
and
, by the results of Section 4, we can easily find the coordinates of
. We have
Theorem 5.1. Let
,
,
, and
such that
. Suppose
. Then
(6)
Proof. From Theorem 4.2, we have
.
We derive the formula (6) by substituting the coordinates of
,
and
. □
Corollary 5.1. Let
,
and
such that
. Suppose
. If
or π, then
(7)
Proof. If
or π, then
. The result can be obtained by the coordinate formula (6). □
It is readily seen that the above result deduced from the coordinate formula (6) is quite equal to that deduced from Equation (3). If
, we have
Theorem 5.2. Let
and
such that
. Suppose
. If
, then
(8)
Proof. According to Theorem 4.3, we have
.
□
For right indefinite dot quotients, we have similar results:
Theorem 5.3. Let
,
,
, and
such that
. Suppose
. Then
(9)
Proof. From Theorem 4.2, we have
.
Then, we derive Formula (9) by the coordinates of
,
and
. □
Corollary 5.2. Let
,
, and
such that
. Suppose
. If
or π, then
(10)
Proof. If
or π, then
. The result holds from the coordinate formula (9). □
Theorem 5.4. Let
and
such that
. Suppose
. If
, then
(11)
Proof. According to Theorem 4.3, we have
.
□
Next, we will discuss the applications of coordinate formulas. Although we can give some application examples of dot quotient in differential manifolds, in physics, in force, etc., here we just give a very simple examples to show how to use our formulas, and verify the correctness of our theory, by the way.
Example 1 It is given that two vectors
and
. Then their dot product
, and their cross product
. And let
. Of course,
and
. Thus we have
,
,
. Since
and
are known, the angle
between them is determined by
. By the way, we have
,
, and
.
It is no doubt that
and
and
. From Formula (6), we, of course, have the coordinates of
:
;
;
.
It is readily seen that
is exactly equal to
.
Similarly, by Formula (9), we can obtain the coordinates of
:
;
;
.
It is also seen that
is really equal to
.
Example 2 Given two vectors
and
, their dot product
, which means
, and
, and their cross product
. If we regard
as known and let
, then
.
Thus, from Formula (8), we have the coordinates of
:
;
;
.
It is readily seen that
is exactly equal to
.
Similarly, we have the coordinates of
:
;
;
.
It is also seen that
is fully equal to
.
We should note that, if we adjust the value of
, we can get different
or
such that
. For instance, if
instead of 3, we have
and
such that
and
. Based on our need, by adjusting the value of
, we can get better or best
or
.
Generally speaking, if we know sufficient supporting information, we really can back to find
(
) such that
from the main information c and
(
). Sometimes, we are not interested in finding original
(
), but in
finding a needed
(
) such that
(
). When
, the
dot quotient theory tells that it is enough by adjusting angle parameter and normal vector parameter, otherwise we need additional conditions such as cross products.
6. Conclusions
This paper successfully set up the theory of indefinite dot quotients by adding angle and normal direction parameters.
First of all, we successfully define indefinite dot quotients by Definition 2.1. By the definition, as we know a real number c and a nonzero vector
, we
inversely obtain two vectors
and
satisfying
where
is an angle parameter and
direction parameter such that
, and both
and
have the same direction, and both
and
have the same direction. We further obtain the basic properties (2.1) to (2.6) and some expected operation properties.
Secondly, we obtained two geometric expressions of indefinite dot quotients (by Theorem 4.1 and Theorem 4.5) where one of them exposes that, when normal direction
is fixed, the end points of indefinite dot quotients form two parallel lines with the change of angle parameters, and another reveals that when
is fixed, the end points form two circles with the change of normal directions. By the geometric expressions, Corollary 4.6 not only puts angle parameter into real parameter but also presents two unified expressions:
which successfully avoid concerning the angle parameter is
or not. We say that the structures of indefinite dot quotients are exposed completely in geometry.
Finally, we obtained the coordinate formulas (6)-(11), which help us to get the
coordinates of
and
easily. With determined angle and direction
parameters, we inversely find the exact vector
(
) from a scalar c and a vector
(
) such that
by the coordinations.
We want to say, we can design new indefinite dot quotients by adjusting angle parameters and direction parameters to fit new situation in the applications.
The relation between dot products and indefinite dot quotients likes that between derivatives and indefinite integrals, the only difference is that an indefinite integral has only one parameter (formed by an arbitrary constant), but an indefinite dot quotient has a parameter pair (an angle, a direction).
It is seen that this paper has successfully built the theory of indefinite dot quotients which solve the long time problem that dot product has no corresponding division in three dimensional space. Our theory of indefinite dot quotients makes the theory of vector analysis more perfect.