Cross Product Calculator

Result = A X B
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Cross Product Calculator

Cross Product Calculator is used to calculate cross Product of two arbitrary vectors. Calculating vector cross product manually is a time consuming process and you might not get exact result sometimes. But our cross product calculator helps you to calculate cross product two vectors in seconds and you will get the exact result. We are offering Vector cross product calculator for free. Cross product calculator also called as vector multiplication calculator is very useful for students, mathematicians, and many others. Cross Product Calculator is one of the best vector calculators available on internet.

What is a cross product ?

Cross product (known as vector product) is a binary operation on two vectors which exist in a three-dimensional space. It is denoted by symbol ✕. For this example we can consider two linearly independent vectors A and B, the cross product  A ✕ B is a vector, perpendicular to both A and B and thus normal to the plane containing them. If two vectors are not linearly independent i.e, they have the same direction or have the exact opposite direction from one another then their cross product is zero. Therefore  the magnitude of the product of the two perpendicular vectors is the product of their lengths.

Vector Cross Product Calculator :

Without a vector cross product calculator it is hard to calculate the cross product of 3D vectors . This calculator can be used to calculate the 3D vectors by using two arbitrary vectors in cross product form.  give you just have to simply type the input values And you will get the cross product directly.  Using this calculator you won’t need to calculate the cross products manually.

How to calculate cross product manually ?

For example let us consider two vectors a and b as :

Cross product calculator

 to remember the results in an easy way rewrite the result as a determinant.

This begins with expanding the product a cross b:

Cross product calculator

 and then we can calculate all the cross products of the unit vectors

Cross product calculator

using determinants we can write the result as the following :

Cross product calculator

The above formula for the 3 x 3 determinant looks like Expansion of 3  x  3 determinant. This can be an easy way to remember the cross product or how to compute cross product .

Cross product calculator

Matrix Cross product

The cross product a × b is defined as a vector c perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude which is equal to the area of the parallelogram that the vectors span.

Properties of cross product

  • The length of the cross product of two vectors is given by :
vector cross product calculator
  • This length of the cross product of two vectors is equal to the area of the parallelogram determined by the two vectors:
vector cross product calculator
  • Anti Commutativity:
Cross product calculator
  • Multiplication by scalars:
vector multiplication calculator
  • Distributivity:
vector multiplication calculator
  • Scalar triple product of vectors (abc):
vector multiplication calculator

Vector cross product

A vector has only direction and magnitude. They can be combined to form a vector cross product in mathematics the vector product and cross product is the combination of the two binary vectors in 3D space and calculated using cross multiply calculator.

The result for the length of the cross product leads us directly to the certainty that two vectors are parallel if and only if their cross product is a 0 vector. This is true since two vectors are parallel to each other if and only if the angle between them is 0 degree or 180 degree.

Vector multiplication calculator Cross Product Calculator

 A vector quantity is a quantity that has both magnitude or size and a direction. These properties must be given in order to define a vector.

A cartesian cross product calculator or a vector calculator is used to solve arithmetic operations to find the magnitude of the vector, whether it is a 2d or 3d vector it works in both cases. It also helps us to calculate the scalar vector of the multiplication. It is also known as vector multiplication calculator.

Formula :

c= a x b = | a | * |b| * sinΘ *n

  • a and b are arbitrary vectors,
  • | a | and | b | are the magnitudes of the above vectors,
  • C is the resulting vector cross product,
  • Θ is the angle between these vectors, and
  • n is a unit vector which is perpendicular to the plane determined by a and b. Its direction can be known by using the right-hand rule.

Some Basic Terminologies for cross product calculator :

  • Orthogonal Vectors:

When you take the cross product(vector multiplication calculator) of two vectors a and b, the resultant vector, (a x b), will be orthogonal to BOTH a and b.  For knowing the direction of a✕b we can use the right hand thumb rule.

  • Parallel Vectors:

Only two non-zero vectors a and b can be parallel if and only if, a x b = 0

Cross Product Properties

If a, b, and c are vectors and c is a scalar, then

1. a x b = –b x a

2. (ca) x b = c(a x b) = a x (cb)

3. a x (b + c) = a x b + a x c

4. (a + b) x c = a x c + b x c

5. a · (b x c) = (a x b) · c

6. a x (b x c) = (a · c)b – (a · b)c

Cross Product of vectors :

Note : The result is a vector and is NOT a scalar value. For this reason, it is also known as the vector product.

  • To make this definition easier to remember, we usually use determinants to calculate the cross product(use vector cross product calculator ).
  • If the angle between the two vectors a and b is θ, then |a x b| = |a||b| sin θ
  • Always remember that the length of a cross product is equal to the area of the parallelogram which is determined by a and b.
  • Volume of the parallelepiped which is determined by vectors, is the magnitude of their triple scalar product as shown below:

V = |a · (b x c)|

  • If the triple scalar product is 0, then the vectors must lie in the same plane which means that they are coplanar.

Hope Our tool Cross Product Calculator solves all your cross product calculations. Soon we are launching few more mathematical calculators to make your work easy. If you like our vector cross product calculator feel free to give a review. Thank you 🙂