Vector product of two vectors: Definition, calculation and applications

The vector product is a mathematical operation between two vectors, represented as a × b. Its result is another vector, with magnitude calculated by the product of the magnitudes of the vectors and the sine of the angle they form. The direction is perpendicular to the plane defined by the vectors.

It is not commutative and can be expressed analytically using Cartesian components or as a module and unit vector. It also has applications in force and magnetic force calculations. Practical examples illustrate its usefulness.

The vector product is a mathematical operation performed between two vectors, represented by a→ and b→, and is denoted as a→ × b→. Its result is another vector, represented by r→. Unlike other mathematical operations, the vector product is not done simply by multiplying the magnitudes of the vectors involved.

The magnitude of the vector product is calculated by multiplying the magnitudes of vectors a and b by the sine of the angle they form. This means that the result of the vector product is proportional to the area of ​​the parallelogram formed by vectors a and b in a three-dimensional plane.

The direction of the vector product is perpendicular to the plane defined by the two vectors. This means that the resulting vector will be perpendicular to both a→ and b→. Additionally, the direction of the vector product can be determined using the right-hand rule. In practice, this means that if the right hand is placed with the fingers extended in the direction of a→ and flexed toward b→, the thumb will point in the direction of the vector product r→.

It is important to note that the vector product is not commutative, meaning that a→ × b→ is not equal to b→ × a→. Depending on the order in which the vector product is performed, the result will be a vector in different directions and directions.

The calculation of the vector product in three dimensions can be done in different ways. Below are two common methods to calculate this mathematical operation.

One way to calculate the vector product in three dimensions is by using the Cartesian components of the vectors involved. Suppose we have two vectors a→ = (ax, ay, az) and b→ = (bx, by, bz).

To obtain the result of the vector product a→ × b→, we can apply the following formula:

Another method to calculate the vector product in three dimensions is using 3 x 3 rank determinants. Taking as an example the same vectors a→ and b→ mentioned above, the result is obtained by applying the following formula:

r→ = | i j k | | ax ay az | | bx by bz |

Where i, j and k are the versors of the x, y, and z axes respectively. By calculating the determinant, the result vector r→ will be obtained.

These methods are fundamental to solve problems involving the vector product in three dimensions. They can be applied in both geometric and physical contexts, allowing crucial information to be obtained about forces, magnetic fields and other vector magnitudes.

The vector product has some important properties that we must take into account when working with it. Next, we will explore two of these properties: associativity and distributivity, and the noncommutativity of the vector product.

One of the fundamental properties of the vector product is its associative property, which states that the result of performing the vector product of three consecutive vectors will be the same, regardless of how they are grouped. This can be expressed mathematically as follows:

On the other hand, the vector product also meets the distributive property with respect to addition. This means that the vector product of two vectors added together is equal to the sum of the vector products of each vector separately. The distributive property can be expressed as follows:

Another fundamental property of the vector product is its non-commutativity. This means that the order in which the vector product is performed will alter the final result. It is important to keep this in mind when using this mathematical operation. Mathematically, it can be expressed as follows:

This property tells us that, in general, the vector product of two vectors is not equal to the vector product of the same vectors in reverse order.

The vector product has an important geometric interpretation that allows us to better understand its operation and applications. In this section, we will explore two fundamental aspects: the area of ​​the parallelogram formed by the vectors involved and the construction of a vector perpendicular to a plane.

One of the most notable properties of the vector product is its relationship with the area of ​​a parallelogram formed by the two vectors. By calculating the module of the vector product between both vectors, we will obtain the area of ​​said parallelogram. This property is very useful in geometry and area calculations in three-dimensional space.

Another relevant application of the vector product is the construction of a vector perpendicular to a plane from two vectors contained in that plane. If we consider two vectors in the plane, the vector product between them will result in a vector perpendicular to the plane defined by said vectors.

The vector product has various applications in different fields of physics and mathematics. Among the most relevant are the calculation of the force torque and the calculation of the magnetic force.

The vector product is used to determine the torque acting on a rotating object. The force torque is the result of the cross multiplication of the distance from the axis of rotation to the point of application of the force and the force vector. It is expressed as a vector perpendicular to the plane defined by the direction of the force vector and the distance to the axis of rotation.

This calculation is fundamental in engineering and mechanics to analyze and predict the behavior of rotating systems, such as engines, heavy machinery, and power transmission devices. It allows us to understand how turns and movements will be generated in these systems and facilitates the design of efficient and safe mechanisms.

The vector product also plays a crucial role in calculating the magnetic force acting on a moving charge in the presence of a magnetic field. Using the vector product between the velocity vector and the magnetic field vector, the direction of the resulting magnetic force can be determined.

This calculation is of utmost importance in particle physics and in electromagnetism applications, such as the generation of electric current in motors and generators. In addition, it is fundamental in sensor and actuator technology, where knowledge of magnetic force allows controlling and manipulating electrical charges in devices such as accelerometers, digital cameras and electromagnets.

The vector product has various applications in the field of physics and geometry. Below, some practical examples will be presented that illustrate its usefulness in different situations.

Suppose we have two forces, F1 and F2, acting on an object in a plane. To calculate the torque moment generated by these two forces, we can use the vector product. By taking the vectors that represent the forces and calculating their vector product, we will obtain another vector that will give us information about the magnitude and direction of the resulting force couple.

In the field of electromagnetics, the vector product is used to calculate the magnetic force exerted on a moving charge. If we know the velocity vector of the charge and the magnetic field vector at the point where it is located, we can obtain the magnetic force vector through the vector product. This will allow us to determine the magnitude and direction of the magnetic force acting on the charge.

The vector product also has applications in the field of geometry. For example, if we have two vectors in a plane and we want to construct a vector perpendicular to said plane, we can use the vector product. The resulting vector will be perpendicular to the plane defined by the two vectors and will allow us to perform calculations and geometric constructions in three-dimensional space.

Another use of the vector product is the calculation of areas in three-dimensional space. If we have two vectors that form the sides of a parallelogram, we can use the vector product to obtain a vector perpendicular to the parallelogram. The magnitude of this vector will be equal to the area of ​​the parallelogram formed by the two initial vectors.