Vector product

Don't you know how to calculate the vector product of two vectors? Use our online calculator and find out the result automatically, without doing any operations.

Just enter the i, j and k components of the two vectors and click the calculate button to get their vector product. For example, if you are given the vector Vector u= (1, 2, 3) y vector v= (4, 5, 6), the values of each of its components are the ones you must write in the tool. If you do not know how to calculate the vector productread on and we will explain it to you.

What is the vector product?

The vector product of two vectors results in another vector with:

  • Perpendicular direction to the two vectors
  • Direction based on the corkscrew rule or right hand.

Vector product

What does the right hand rule say? The right hand or corkscrew rule tells us the direction that the vector will have. according to the movement of a corkscrew (or screwdriver, screw...etc).

In the case of the figure above you can see that if we turn the corkscrew to the left (counterclockwise), the corkscrew goes upwards and, therefore, that will be the direction of the resulting vector "c".

It is very important that you do not confuse the vector with the scalar product.

How to make the vector product of two vectors

To solve the vector product of two vectors Vector u= (a1, a2, a3) y vector v= (b1, b2, b3), we have to apply the following formula which will give us the components of the vector "c":

Formula for the vector product of two vectors

Basically, it is necessary to write in a 3×3 determinant and decompose it into three 2×2 determinants for each component i, j, k.

You do not need to remember the elements of the determinant associated with each element as they can be easily deduced. Simply you have to calculate the attached determinantthat is, you have to eliminate the row and column in which each element i, j, k is located in such a way that with the remaining elements you form a 2×2 determinant. Here we show you how to solve a determinant of a 2×2 matrix.

For example, we are going to see a solved exercise in which we are asked to calculate the vector product of the vectors  Vector u= (2, 0, -1) y vector v= (1, 1, -2):

Solved exercise of vector product of two vectors

As a result of the vector product, we have this vector:

Result of vector product of two vectors exercise

There is a second method for calculating the vector product of two vectors from the following mathematical formula:

Formula for the vector product of two vectors

The method consists of multiplying their moduli by the sine of the angle they form and the unit vector "n which is orthogonal to the vectors and whose sense and direction is governed by the right hand rule.

Unit Vector Formula

If you don't know how to do it, here is how to do it calculate the modulus of a vector.

Modulus of the vector product

For calculate the modulus of the vector product of two vectors you have to use this formula:

Formula for calculating the modulus of a vector product

In this case, the result will be a scalar number that represents the length that the segment of the resulting vector will have.

Vector product of two vectors in R2

If we are asked to calculate the vector product of two vectors in R2The process to follow is exactly the same as in R3. We will simply have to complete with a 0 the component of the vector that we are missing.

For example, let's calculate the vector product of the vectors in R2 Vector u= (3, 2) y vector v= (2, -1).

As you can see, we have no "k" component, so the elements a3 y b3 will be zero. Taking this detail into account, we solve the exercise and we are left with the following:

Example of vector product of two vectors in R2

Triple vector product

The triple vector product (also known as double vector product) consists of multiplying two vectors vectorially and with the result obtained make another vector product with the missing vector. For example:

A × (B × C) = B (A - C) - C (A - B)

By multiplying the vectors B x C we generate a vector that we then have to multiply vectorially by A. The result would be a vector contained in the plane defined by B and C.

As the double vector product has an anticommutative propertyWe can also express it in this way:

A × (B × C) = - C x (A x B)

Finally, it should be noted that the triple product has no associative property.

Properties of the vector product

Here are the properties of the vector product that have not yet been mentioned:

  • Anticommutative: a x b = - (b x a)
  • a - (a x b) = 0
  • If a x b = 0 being a≠0 and b≠0, this means that the vectors are parallel and so their vector product is zero.
  • (a + b) x c = a x c + b x c
  • a x (b x c) + c x (a x b) + b x (c x a) = 0 according to the Jacobi identity

Vector product applications

We have many applications for the vector product in the world of mathematics, physics or astronomy.

Some of those that are always used as examples and which allow us to calculate the area and volume of some figures geometric shapes such as a triangle or a parallelogram.

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