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The properties of the cross product

The definition of the cross product

In fact, modern mathematics, doubtlessly, appertains to the class of scientific disciplines that are widely used in different fields of study, as well as in various spheres of mundane life. Nowadays, even a simple assignment, such as formatting an essay or checking sports competition results often requires knowledge of modern vector analysis. Furthermore, even the simplest completely automatic thesis generator operates with the values of variables, according to the principles of tensor and vector analyses. Thereby, the importance of this topic for all undergraduates can be hardly overestimated. Thus, let us examine one of the fundamental concepts of the vector analysis – the cross product.

The cross product is a pseudo perpendicular to the plane, built in two multipliers, which is the result of a binary operation “vector multiplication” on vectors in the three-dimensional Euclidean space. Naturally, the definition of the cross product can be given in different ways in various exemplars of mathematical enchiridions or historical directories. The cross product is neither commutative nor associative (in fact, it is anticommutative) and differs from the scalar product of vectors. The importance of the cross product, which is, doubtlessly, may seem quite mysterious and over sophisticated for beginners in the sphere of pure math, is expressed in the fact that in different fields of study (both theoretical and practical) there exist many problems about determination of the vector that is perpendicular to the two existing. In fact, this theoretical assignment has great practical value in various fields of engineering, architecture, mechanical engineering and mechanics. The concept of the ‘cross product ’ supplies us is with this possibility. In addition, the cross product is extremely useful for ‘measuring’ perpendicular to the concrete vector. Hence, the length of the cross product of two different vectors, if they are perpendicular, is always equal to the product of their lengths and is reduced to zero, if the vectors are parallel or antiparallel. Determination of the cross product can be different, according to the number of dimensions in which we carry out our computations, and theoretically, in a space of arbitrary dimension ‘n’ we can calculate the cross product of the n-1 vectors while receiving only a vector perpendicular to all of them. Nevertheless, if the cross product is often used in order to limit the non-trivial binary vector, the traditional cross product is defined only in the three- and seven-dimensional spaces. Therefore, we can postulate that the result of the cross product as a scalar depends on the metric of the Euclidean space.

In contrast to the well-known formula, which determines the fundamental principles that allow us to calculate the coordinates of the scalar product of vectors in the three-dimensional Cartesian coordinate system, the formula for the cross product can be determined in accordance with the orientation of the rectangular coordinate system or, alternatively, its ‘chirality’. In fact, the cross product of a vector space is a vector that satisfies the following requirements:

  • The length of the vector is equal to the product of the lengths of the vectors and the sine of the angle between them.
  • The vector is orthogonal to each of the vectors.
  • The vector is directed in such a way that the triple of vectors is ‘right’.

Of course, this concise list represents only the basic properties of the cross product. With an eye to find more specific data about different important nuances of this, undoubtedly interesting mathematical concept, one has to contact a highly specialized college paper writing service, which supplies undergraduates with concrete specialized information about some properties of the cross product, such as its associative property or the distributive property.

The cross product: the main geometrical and algebraic properties

Firstly, a necessary and sufficient condition of collinearity of two nonzero vectors is the fact that the cross product of these two vectors is equal to zero. The magnitude of the cross product, obviously, depends on the sinus of the angle between the original vectors. In other words, the concept of the ‘cross product ’ can be expounded as the degree of two different ‘perpendicular’ vectors. In addition, it is obvious that in this case the inner product can be considered as the degree of ‘parallelism’. If the initial vectors are perpendicular than the cross product of two unit vectors is 1 (unit vector). The cross product is equal to 0 (zero-vector), if the vectors are parallel or antiparallel.

Two vectors are collinear if and only if their cross product is equal to the zero vector. If the vectors are collinear, their cross product is a zero-vector according to our definition. Let us prove the sufficiency. If axb = 0, | axb | = 0; | a | | b | sin φ = 0, where φ – the angle between the vectors a and b. Therefore: for at least one of the three equations: | a | = 0, | b | = 0 or sin φ = 0. Each of these equations implies the collinearity of the vectors a and b. If the vectors a and b are not collinear, the module | axb | of their cross product is equal to the area of the parallelogram constructed on these vectors as adjacent sides.

Lets us postulate the most important properties of the cross product. Firstly, the cross product is anticommutative. Therefore: axb = -bxa. Secondly, the cross product has the specific associative property of multiplication. Hereby: (λa) xb = λ (axb). Thirdly, the cross product has the distributive property with respect to addition (a + b) xc = axc + bxc.

In order to prove the first property (anticommutativity), we note that if the vectors a and b are collinear, then on both sides of axb = -bxa there is the zero-vector. If the vectors a and b are not collinear, then there exists a plane, in which they are parallel. We know that according to the first condition of the cross product, vectors axb and bxa are perpendicular to this plane and, therefore, they are collinear. It is clear that the length of the vectors axb and bxa are as same as the area of the same parallelogram. It remains to prove that the vectors axb and bxa have the opposite direction. This statement follows from the fact that if the triple vectors a, b, axb is right, then the triple b, a, axb is left. Therefore, replacing the last three of the third vector to the other, we get the right three vectors b, a, -axb. It also should be noted that the vector -axb is collinear with the vector bxa and has the same length. According to our definition, it means that the vector -axb is the cross product of the vectors b and a. Therefore, according to our computations: axb = -bxa.

Our calculations for the second property of the cross product (associative property) are practically analogical. In the case of collinear vectors a and b, and according to the condition that λ is equal to zero the vectors (λa), xb and λ (axb) are equal to the zero vector. Furthermore, each of them is either a cross product of collinear vectors, or the cross product of a vector that is multiplied by a number, which is equal to zero. Therefore, in these cases, equality (λa) xb = λ (axb) is also satisfied.

Now, let us suppose that the vectors a and b are non-collinear, and λ = 0. We can show that the left and right sides of this equation are collinear vectors of equal length. Indeed, if we assume that the vectors a, b and λa have a common origin, the pair a, b and λa, b non-collinear vectors generate the same plane, which is perpendicular to their cross product axb and (λa) xb. Therefore, the vectors λ (axb) and (λa) xb collinear. After a series of calculations that are aimed at determination of their lengths, we can demonstrate that these lengths are equal. Two collinear vectors of equal length, either coincide or are opposed to each other. We exclude quite a final opportunity, proving that the vectors (λa) xb and λ (axb) are unidirectional. Obviously, if λ> 0, then the vectors a and λa are unidirectional. Consequently, the vectors (λa) xb and axb also are unidirectional. Moreover, if the vectors axb, λ (axb) are unidirectional, then the vectors (λa) xb and λ (axb) are also unidirectional. If λ <0, then the vectors a and λa are oppositely directed. Consequently, the vectors (λa) xb and axb also are oppositely directed. Multiplying the vector by a negative number axb λ changes its direction to the opposite. Therefore, the vectors (λa) xb and λ (axb) have the same direction.

The cross product and the quaternions: historical references

According to the variety of modern historical encyclopedias that are dedicated to the history of vector analysis and different dissertation editing services the invention of the concept of the ‘cross product ’ was made in 1846 by William Hamilton. In truth, William Hamilton came to this discovery, which largely determined the structure of modern mathematics, algebra, tensor analysis and vector analysis, through the concept of ‘quaternions’. The quaternions are hypercomplex numbers of the form w + ix + jy + kz. In this case w, x, y, and z represent the real numbers and unit vectors, directed along the x, y, and z-axes respectively, are represented as i, j and k. The t, j, and k units obey the following principles: ij = k; jk = i; ki=j; ji = —k; kj = — I; ik = —j;

Therefore: ii = jj = kk = -1;

It is to be noted that for two quaternions q and q’, qq’ does not in general equal q’q. The loss of commutativity in quaternions, while it is very important historically, is also significant mathematically, because this complicates computations in which quaternions are used. Thereby, it may be verified that quaternion multiplication is associative. Hamilton believed that quaternions represented the mathematics of the future and consequently. In fact, this is the reason why he devoted more than twenty years of his life to this problem. The Hamilton’s annotated bibliography proves that he published 109 articles about quaternions by the end of 1865. That is more than 73% of all papers, which were dedicated to this theme. Unfortunately, this document does not supply us with complete information about his drafts. Those who wish to know better what is an annotated bibliography that is dedicated to the specific mathematical theme have to understand that this document does not comprise information about works that have not been published by the author. However, the result of these works was not quite satisfying from the point of view of the modern scientist. The standpoint of quaternions held by virtually all modern mathematicians of the present, however, significantly varies. The contemporary consensus is that the quaternion system is but one of many comparable mathematical systems. Of course, it is interesting as a rather special system, but it offers small value for practical application. In fact, even in the period immediately after their discovery quaternions were strongly criticized because of the abandonment of the commutative property for multiplication. Nevertheless, from the contemporary standpoint it is quite obvious that Hamilton’s discovery was epoch making, for quaternions were the first widely known significant number system that did not obey the standard well-known principles of ordinary arithmetic. Moreover, it was Hamilton who introduced the term scalar (and also the term vector) in its precise mathematical sense, although the similar term radius vector had been used for many years before.

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