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There are two methods of multiplication of vectors by other vectors, in addition to the conventional addition and multiplication of vectors by scalars.

The dot product, for example, is a scalar product; the dot product of two vectors produces a scalar quantity.

The cross product, on the other hand, is a vector product since it produces instead of a scalar another vector.

In this article we would learn what is a cross product and how we can integrate two integrations.

Cross Product

As mentioned above another type of perpendicular vector to two vectors other than dot product is the cross product of the two vectors.

In the disciplines of physics, engineering, vector calculus, and linear algebra the cross product and its formula is applied.

However, when employing the cross product, there are a few things to keep in mind.

  • Two vectors product a × b is perpendicular to u and v
  • The right hand rule is used to find the direction of the vector
  • To find the length of u × v following expression is used
a × b = |a| . |b| sinΘn

Moreover, the determinant is intrinsically tied to the cross product. The cross product, in instance, is not a determinant since a determinant is a number, not a vector.

However, the determinant helps us as a handy memory tool for computing the cross product.


The Cross Product u × v generates a new vector that is perpendicular to both u and v when given two vectors u, v ∈ R3.

Although this may appear to be the simplest definition at first, its valuable features will quickly become apparent.

Using the principles of determinants, it is simple to recall the formula for the cross product.

Finding the Cross Product of two Vectors

The cross product calculation equation is quite easy and straightforward.

The magnitude of vector a multiplied by the magnitude of vector b multiplied by the sine of the angle between them will be the cross product between vectors a and b. Thus, the two vectors

a= < a1, a2, a3>


b = < b1, b2, b3>

cross product is given as

a̅ × b̅ = < a2b3 - a3b3, a3b1 - a1 b3, a1b2 - a2 b1 >

You can also use a cross product calculator for finding answers.

The Right Hand Rule

The right hand rule is utilized, however, to determine the direction of the resulting vector.

The vectors are represented by the thumb (u) and index finger (v) held perpendicular to one another, while the direction of the cross vector is shown by the middle finger held perpendicular to the index finger and thumb.

Vector Integration

Two vectors could be integrated using the same approach, utilized to integrate scalar functions and parametric functions with vector valued functions.

When integrating a vector-valued function, the horizontal and vertical components are integrated individually.

The integration will result in a new vector-valued function or a new vector when you calculate a certain integral.

Let us see comprehensively how you can integrate the vectors. Remember from the calculus of the single variable, the F function that fulfils F′(x)= f(x) is an antiderivative to the independent variable’s x function f.

Then the indefinite integral

∫f(x) dx

is defined as the general antiderivative f.

Furthermore, note that the general antiderivative involves the addition of a constant C to show that the general antiderivative is indeed a whole function’s family.

With vector-valued functions we can perform the same thing.

A vector valued function’s antiderivative Rr is an R vector valued function, in such a way that

R′(t) = r(t)

A vector valued function’s (r) indefinite integral ∫r(t) dt is the universal antiderivative of r and reflects all r antiderivatives collected.

The same rationale to distinguish a vector-valued function from according to component also applies to integration.

Note that the integral of a sum is the sum of the integrals and that constant factors can be removed from integrals. It follows, therefore, that we might integrate components with

r(t) = x(t)i + y(t)j + z(t)k.


The reason is to learn about cross products whether you are learning from cross product calculator or integration calculator or by the simple formulas given in this article .

Happy Learning !