Cross Product

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Define Cross Product

The vector product or cross product of two vectors is denoted as \[\vec{A}\] x \[\vec{B}\] for a resultant vector. The resultant vector depicts the cross product which is perpendicular to the plane surface that spans two vectors. In the case of a dot product, the angle placed between the two vectors or the length of two vectors is found. \[\vec{A}\] x \[\vec{B}\] refers to a cross product of two vectors where one is at a right angle to the other and is formed in the three-dimensional plane. 


Cross Product Formula

In Mathematics, the cross product is defined as a binary function performed on two vectors in a three-dimensional plane. It portrays two vectors that are perpendicular to each other. The cross-product meaning relies on the concept of vectors. The cross - product definition in Maths is related to the product of a vector is denoted as a x b.

The resultant vector obtained can also be demonstrated using the Right-hand rule.

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  1. The Cross-multiplication Formula  

This follows the cross-multiplication method formula to find a solution for a pair of linear equations.  

If two linear equations are placed as a\[_{1}\]x + b\[_{1}\]y + c\[_{1}\] = 0 and a\[_{2}\]x + b\[_{2}\]x + c\[_{2}\] = 0 then the value of x and y can be formed using this method.


  1. Vector Triple Product Formula 

This uses the values of u, v, and w. Consider the formula, u x (v x w). u x v x w ≠ u x v x w.

Note that (u x v) x w is perpendicular to u x v. This is the normal plane is determined by u and v.


  1. Cross Price Elasticity Formula  

The Cross elasticity (Exy) determines the relationship between the two products of a vector. The sensitivity of quantity demand change of product X to a change in the price of product Y is found. Price elasticity formula: 

E\[_{xy}\] = \[\frac{\text{Percentage change in quantity demanded of X}}{\text{Percentage change in the price of Y}}\]


  1. Cross Product of Two Vectors Formula  

for this formula, we consider the following set-up,

If θ is the angle between the given vectors, the formula for the cross product of vectors is given by

A x B = AB sinθ

Cross product of two vectors is indicated as:

\[\vec{X}\] x \[\vec{Y}\] = |\[\vec{X}\]| . |\[\vec{Y}\]| sin θ

\[\vec{X}\] = x\[\vec{i}\] + y\[\vec{j}\] + z\[\vec{k}\]  

\[\vec{Y}\] = a\[\vec{i}\] + b\[\vec{j}\] + c\[\vec{k}\]   

\[\vec{X}\] x \[\vec{Y}\] = \[\vec{i}\](yc - zb) - \[\vec{j}\](xc - za) + \[\vec{k}\](xb - ya)    

 

Properties of Cross Product

The properties of a cross product can vary depending on the type of cross-product formula that is used.


1. General Properties of a Cross Product

  • Length of two vectors to form a cross product: 丨\[\overset{r}{a}\] x \[\overset{I}{b}\]丨 = 丨a丨丨b丨sin θ 

  • This length is equal to a parallelogram determined by two vectors: 

  • Anti-commutativity : \[\overset{r}{a}\] x \[\overset{I}{b}\] = -\[\overset{I}{b}\] x \[\overset{r}{a}\]  

  • Scalar multiplication: (c\[\overset{r}{a}\]) x \[\overset{I}{b}\] = c(\[\overset{r}{a}\] x \[\overset{I}{b}\]) = \[\overset{r}{a}\] x (c\[\overset{I}{b}\])

  • Distributivity: \[\overset{r}{a}\] x (\[\overset{I}{b}\] + \[\overset{r}{c}\]) = \[\overset{r}{a}\] x \[\overset{I}{b}\] + \[\overset{r}{a}\] x \[\overset{r}{c}\] 

2.Vector Triple Product Properties

  • The vector quantity is the vector triple product - \[\overset{r}{a}\].(\[\overset{I}{b}\] x \[\overset{r}{c}\]) = (\[\overset{r}{a}\] x \[\overset{I}{b}\]) . \[\overset{r}{c}\])

  • Unit vector coplanar with a and b is perpendicular to c. 

  • The vector triple product is often used in rotational studies in Physics.

3. Properties of The Scalar Triple Product

  • When the vectors are cyclically permuted, then \[\overset{I}{a}\] x (\[\overset{I}{b}\] x \[\overset{r}{c}\]) = (\[\overset{r}{a}\].\[\overset{r}{c}\])\[\overset{I}{b}\] - (\[\overset{r}{a}\].\[\overset{I}{b}\])\[\overset{r}{c}\]

  • The product of two vectors is cyclic

  • When a triple product is zero, this can be inferred as vectors in coplanar nature.

  • Zero arises when three vectors have zero magnitudes.

  • Using the scalar triple product, the volume of a given parallelepiped vector is obtained.

If the triple product of vectors is zero, then it can be inferred that the vectors are coplanar.

 

The volume of a parallelepiped is indicated by a triple product vector. If it is zero, any one of the three vectors is found and exhibits zero magnitudes. The vectors a and b can be indicated by its perpendicular position to the plane. The dot product of the resultant with c will only be zero if the vector c also lies in the same plane.

 

Calculating Cross Product

To calculate the cross product for a given set of vector equations, be sure to pay attention to the planes they reside in and the equations provided.

Let us look at the following example to strengthen our basics in this concept,

Find the cross product for the given vectors: \[\overset{\omega}{X}\] = 5\[\overset{I}{i}\] + 6\[\overset{I}{j}\] + 2\[\overset{I}{k}\] and \[\overset{\omega}{Y}\] = \[\overset{I}{i}\] + \[\overset{I}{j}\] + \[\overset{I}{k}\]

We know that,

\[\overset{I}{X}\] = 5\[\overset{I}{i}\] + 6\[\overset{I}{j}\] + 2\[\overset{I}{k}\]

 \[\overset{r}{Y}\] = \[\overset{r}{i}\] + \[\overset{r}{j}\] + \[\overset{r}{k}\]

To obtain the cross product, vectors are written in the determinant form. 

\[\overset{I}{X}\] x \[\overset{I}{Y}\] = (6 - 2)\[\overset{I}{i}\] - (5 - 2)\[\overset{I}{j}\] + (5 - 6)\[\overset{I}{k}\]     

Therefore, \[\overset{I}{X}\] x \[\overset{I}{Y}\] = 4\[\overset{I}{i}\] - 3\[\overset{I}{j}\] - \[\overset{I}{k}\] 

FAQ (Frequently Asked Questions)

1.What are the Four General Properties of a Cross Product?

Ans. Cross products are a fundamental aspect of Mathematics. The cross product is dependent on the plane and direction the vectors are placed in. The cross product for any two vectors relies on specific properties to obtain a cross-product result. The properties like anti-commutative property and zero vector property plays a pivotal role in seeking the cross product of two vectors. 

 

Some other secondary properties that are also applied are Jacobi property and distributive property. The properties of cross-product are given below:

  1. Anti-communicative Property - A x B = -B x A  

  2. Distributive Property -  A x (B x C) = A x B + A x C 

  3. Jacobi Property - Ax (B x C) + B x(C x A) + C x(A x B) = 0  

  4. Zero Vector Property - a x b = 0 when a = 0 or b = 0.

 2. What are the Applications of a Cross product?

Ans. The applications of a cross product are as follows:

  • The cross product of vectors has established a variety of applications in engineering and mathematical streams.

  • The primary application of a cross product is in dealing with rotating bodies in astrology and physics. 

  • It is also used to find the vector perpendicular concerning other vectors provided. 

  • It gives a sense of direction, magnitude, and sometimes speeds of the object set in motion. 

  • They are also used in calculus along with other formulas. Cross products can produce determinants.

  • They describe the force of an object and its direction.

  • Sometimes, the direction of the gravitational field can also be devised using a cross product of two vectors