Ever heard about vector algebra? Well! It is a branch of algebra that deals with the vector and scalar quantities and their operations, like addition, multiplication etc. The most significant is the cross product and dot product.
Both involve multiplication of two vectors, but the difference lies in the fact that a cross product or vector product always results in a vector quantity. On the other hand, a dot product always results in a scalar quantity (having magnitude but no direction).
In this article, we will discuss the vector product its definition, history and real world applications. Now, let’s get started.
Cross product definition:
It is defined as a binary operation on two vectors that are present in a three dimensional space, and the product results in a third vector, which is perpendicular to both vectors and standard to the space containing them.
The image below illustrates the cross product of two vectors a, b that results in a third vector a×b and its direction is perpendicular to the initial vectors.
The introduction of the dot and cross products dates back to the 18th century, for the first time used by an Italian mathematician and astronomer Joseph Louis.
These components were introduced for the study of tetrahedron, a polyhedron composed of four triangular faces. To study it in three dimensions, these two products were used.
Later in 1843, William Rowan invented the quaternion product and also explained the terms vector and scalar. After 40 years of the discovery of quaternion, Josiah Willard Gibbs found that the existing quaternion system was too cumbersome. Because the result of scalar products was to be worked separately.
Here, the cross product and dot product were accepted as standard for vector and scalar geometrical studies.
The vector product has a wide range of practical significance, particularly in the fields of physics and geometry.
Torque and Angular momentum
In mechanics, the torque, is defined as the turning effect or rotational effect of a linear force applied on an object. Whereas, the angular momentum is the same as the rotational effect that is correspondent to linear momentum, denoted by L.
Vector products are used to define these two vector quantities. For instance, the above expression;
L = r × p
Here, L is the resulting vector (angular momentum), r is a position vector in relation to the origin, and p represents the linear momentum.
In the field of computational geometry
The vector product appears in the estimation of the distance and direction of two skew lines (having different planes) this distance is measured in 3d space.
These are used to perform a frequent operation of geometry called computing the normal for a polygon or triangle. For instance, to estimate the curving of a polygon, whether clockwise or anticlockwise. It is measured at some point within the polygon.
The method used to calculate it is called triangulating the polygon, dividing into triangles and then summing up the angles, here we use the vector product to keep track of each angle. These cross products can also be used to measure the volume of a polyhedron.
In the end, I sum up this article, these components of vector algebra, the dot and cross products are quite important operations, in terms of both physics and geometry.
If you are interested in solving problems related to vector products, we suggest you try some online tools that are specifically designed to compute the above mentioned problems. Give cross product calculator a try. Good luck!