Scalar and Vector Products

Introduction

The numbers or magnitude can measure a scalar product whereas the vector products are described with the direction along with the magnitude. A scalar and two vectors can be multiplied and can form two kinds of products according to the rules of physics. Both scalar and vector products are used for defining energy and work performed, especially within energy relations.

Definition of Scalar Products and Vector Products

Scalar products can be defined as an algebraic operation that involves two equal-length sequences of numbers and as a result, returns only one result at a time. This product can be found by taking the component of one vector towards a certain direction of the other vector after multiplying it with the magnitude of the other vectors (Pellicer & Sola-Morales, 2019). Products like distance, mass, volume; energy and length are some examples of scalar products.

Figure 1: Scalar products

In terms of representing the scalar products with mathematical equations, the formula of "$\mathrm{\vec{A}.\vec{B} \:= \:|\vec{A}| |\vec{B}|\: cos \:\phi}$". In this formula,$\mathrm{|\vec{A}|}$ is the magnitude of vector A, $\mathrm{|\vec{B}|}$ is the magnitude of vector B and $\mathrm{\phi}$ is the angle between vector A and vector B. Scalar products are also defined as the dot products of two vectors while explaining scalar quantities in physics. A vector $\mathrm{(^\vec{})}$ can be defined as an object with magnitude along with direction. A vector product can be outlined in the form of an arrow that can denote the length of the magnitude with the direction. A vector generally travels from the trail direction towards the head direction with its magnitude. Characteristically, the vector products are also commonly known as the area product or the cross product due to their joining with three-dimensional space.

Figure 2: Vector products

Vector is commonly denoted by "$\mathrm{\vec{a} \:x \:\vec{b}}$ where the resultant of the product appears perpendicular to the original vectors. In order to demonstrate the vector products, the formula of "$\mathrm{\vec{A} \:x \:\vec{B}\: =\: |\vec{A}||\vec{B}|sinθn}$ is used in physics. In this formula, $\mathrm{|\vec{A}|}$ is the length of the magnitude of vector A, $\mathrm{|\vec{B}|}$ is the magnitude of vector B, θ is the angle between vector A and vector B and finally n is the unit vector perpendicular to vector A and vector B (Hyperphysics, 2022).

Properties of Scalar and Vector products

Scalar products have a few distinctive properties that make them useful in real-time. First, in terms of scalar products, the direction of the angle within the equation does not have any type of significance between two vectors. It can only be measured from one vector to another since $\mathrm{cos\:\theta \: = \: cos \:(-\theta) \: = \: cos\:\theta (2\pi - \theta)}$. The angle within a scalar product appears at more than 90°, less than or equal to the 180° then the product appears with a negative value. So the mathematical equation of the scalar products is $\mathrm{90^{\circ}\:\lt\: \theta \:\lt\: = \:180^{\circ}}$.

Vector products on the other hand, have two significant properties which make them distinctive from the scalar products. First, most vector products are based on the right-hand screw rule for obtaining the proper and required direction in real-time (Sun et al. 2021). Most importantly, the vector products are non-communicative as it is concluded from its mathematical equation of $\mathrm{\vec{b}\:x\:\vec{a}\:=\:-a\:x\:\vec{b}}$

Figure 3: Three-dimensional vector

Another significant property of a vector product is its tendency for distribution over addition. This particular property stands equal with the properties of scalar products, as it can be seen within the dots. Vector also involves a three-dimensional structure based on the gateway of the dot products (Thefactfactor, 2022). Finally, it can be said that a vector product can be altered as it remains equivalent when the direction of the vector changes in real-time.

Difference between Scalar Products and Vector Products

Despite an integral relationship with each other, the scalar products and vector products are quite distinctive. The first difference appears in the representation of the products, where a scalar product is represented with a dot and a vector product is represented with a cross. Further, the relation between the scalar products is defined as "$\mathrm{\vec{A}.\vec{B}\:=\:AB\:Cos\:\theta}$ and the vector products are defined as "$\mathrm{\vec{A}\:\times\:\vec{B}\:=\:AB\:Sin\:\theta}$

Most importantly, the scalar products follow the communicative law whereas the vector products do not obey the communicative law at all. Lastly, the scalar products appears perpendicular to each other with a value of A.B = 0 (Geeksforgeeks, 2022). The vector products on the other hand, appear parallel to each other, with the value of "$\mathrm{\vec{A}x \:\vec{B} = 0}$.

Conclusion

The scalar products can be defined as the multiplication operation of two vectors at a time. The vector products on the other hand, are used for locating a certain point in space concerning another point in the space. Despite their similarities and integral appearance, these products are represented differently with their mathematical equation in real-time.

FAQs

Q1. What is the relationship between Scalar and Vector products?

Ans. The scalar and the vector products are commonly used in engineering and physics, as they both can be used after mathematical multiplication. In a way, the scalar products are based on vectors in real-time so the relationship establishes between them, is based on work and energy.

Q2. What the utilizations are of vectors in real-time?

Ans. Vectors are essential for representing the direction and magnitude through the additional rules. In terms of defining velocity, it plays a key role in outlining the magnitude of speed in real-time.

Q3.How are the Scalar and Vector products represented?

Ans. Based on their distinctive characteristics, scalar products are called dot products and the vector products are called cross products. Therefore the scalar products are represented with a dot (.) and the vector products are represented with a cross (x).