Cross Product

Dhristi JEE 2022-24

What is Cross Product?

The cross product which is also referred to as the vector product of the two vectors can be denoted as A x B for a resultant vector. This resultant vector represents a cross product that is to the plane surface that spans two vectors. In the situation of a dot product, we can find the angle placed between the two vectors. Ax B this arrangement is known as a cross product of the two vectors where one vector is at a right angle to the other and all of these are present in a three-dimensional plane.


Cross product refers to a binary operation on two vectors in three-dimensional Euclidean vector space. The right-hand rule is used to calculate the cross product of two vectors. The right-hand rule is mainly the result of any two vectors which are perpendicular to the other two vectors. The magnitude of the resulting vector can also be calculated using a cross product.


If θ is the angle between the given vectors, then the formula is given by


\[A\times B=AB\sin\theta\]


\[\vec{A}\times \vec{B}=absin\theta\hat{n}\]


Where \[\hat{n}\] is the unit vector.


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 \[a_{2}x+b_{2}y+c_{2}=0\]  then the value of x and y can be formed using this method.


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 normal plane is determined by u and v.


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: 


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


Cross product of two vectors is indicated as:

\[\vec{X}\times \vec{Y}= \vec{\left | X \right |}.\vec{\left | Y \right |}sin\theta \]

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

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

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


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 

\[\left | \vec{a}\times \vec{b} \right |= \left | a \right |\left | b \right |sin\theta\] 

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

Anti-commutativity  \[\vec{a}\times \vec{b} = -\vec{b}\times \vec{a}\]

Scalar multiplication: \[(c\vec{a})\times \vec{b} = c(\vec{a}\times \vec{b})= \vec{a}\times (c\vec{b})\]

Distributivity \[\vec{a}\times \left ( \vec{b} +\vec{c}\right ) = \vec{a}\times \vec{b} + \vec{a} \times \vec{c}\]


2.Vector Triple Product Properties

The vector quantity is the vector triple product - \[\vec{a}.(\vec{b}\times \vec{c}) = (\vec{a}\times \vec{b}).\vec{c}\]

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 \[\vec{a}\times (\vec{b}\times \vec{c}) = (\vec{a}\vec{c})\vec{b} - (\vec{a}\vec{b})\vec{c}\]

  • 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:


\[\vec{X} = 5\hat{i}+ 6\hat{j}+ 2\hat{k}\] and Y = \[\vec{X} = \hat{i}+ \hat{j}+ \hat{k}\]


We know that,


\[\vec{X} = 5\hat{i}+ 6\hat{j}+ 2\hat{k}\]


\[Y = \vec{X} = \hat{i}+ \hat{j}+ \hat{k}\]


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


\[\vec{X}\times \vec{Y} = (6-2)\hat{i}- (5-2)\hat{j} + (5-6)\hat{k}\]


Therefore, \[\vec{X}\times \vec{Y} = 4\hat{i}-3\hat{j}-\hat{k}\]

FAQs on Cross Product

1.  What are the Applications of a Cross product?

The Applications of a Cross Product are as Follows:

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. 

2. How to find the direction of a cross product?

We can find the direction of the cross product of two non zero parallel vectors a and b  by the right hand thumb rule. In your right hand, if you point your index finger along the vector a and point your middle finger along vector b, then the thumb gives the direction of the cross product. 

3. Define Triple Cross Product.

The triple cross is defined as the  product of three vectors . In other words, the cross product of one vector with the cross product of another two vectors.

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