Vector Algebra-Scalar Product of Two-Vectors

In mathematics, most qualities are classified either as scalar or vector quantities. Also referred to as dot product, the scalar product of two vectors is a value calculated by multiplying its vector components. It bears significance only for the value of vectors involved in multiplication that possess the same number of dimensions. However, we must have both direction and magnitude to obtain the product of vector quantities.

In the context of a two-dimensional Cartesian plane, the vector components are denoted with reference to the X and Y coordinates of their endpoints.

What is a vector? 

A vector refers to a quantity that possesses both direction and magnitude in mathematics. However, it does not have any position. These are represented as line segments with an arrow on the one end with their lengths as their magnitudes. These are usually expressed through a boldface alphabet. Its magnitude or length is represented by v, which indicates a single-dimensional quantity known as a scalar.

However, operating multiplication on a vector by a scalar changes its overall length without impacting its direction. It can be used to calculate the scalar product of two vectors as required by the situation. Upon multiplying a vector scalar by a negative value reverses the direction of its arrow.

For instance, if we multiply a vector by ½, it will reduce its length to its half as long in the same direction. On the other hand, if we multiply the same vector by -3, it will double the length of the vector, but it will be pointed towards its opposite direction.

Scalar Product of Two-Vectors

Now that we have the basic idea of what a vector means let us answer our question: What is the scalar product of two vectors?

Also known as dot product, scalar product refers to the multiplication of two vectors which results in a scalar value. This method applies only to the pair of vectors that hold the same value in dimensions. The scalar product of two vectors can also be obtained by performing a specific operation on vector components that are not some.

There are two ways to define their representations. These are explained below:

Algebraic Expression

In algebraic terms, the value that defines the vector space is expressed in terms of its product. Using this approach, the dot product is derived from the sum of the products with respect to the entries of two subsequent numbers. In this case, the first vector a will be equal to [a1, a2, …, an], and the second vector b will be equal to [b1, b2, …, bn].

This will be given by: a.b = (a1b1 + a2b2 + a3b3)

Further, the scalar product can be written as a matrix product if the vectors are defined as row matrices. However, it is essential to note that it is just another way of representing the values in algebraic terms.

Geometric Expression

Euclid’s theory derives the geometric explanation of the scalar product of two vectors. According to the Geometric explanation, the cosine of the angle between two vectors is multiplied by its magnitude values. As per this representation, the scalar product between two vectors, let us say a and b, will be represented as: a.b = |a||b| cos θ.

Here, θ denotes the angle between the vectors a and b while |a| and |b| refer to the magnitudes of the two vectors.

In this case, if the angle between these two vectors is 90, a.b would be 0. On the other hand, if the two vectors run parallel to each other, then a.b = |a||b|

Scalar Product Properties of Vector

The scalar product of two vectors functions in respect of the following properties if the values of a, b and c identify as real vectors and r has a scalar value:

• Commutative – The commutative property of a vector implies that a.b = b.a. This equation further follows that a.b = b.a = ab cos θ.
• Distributive – As per the distributive property of the scalar product of two vectors, a.(b+c) = a.b + a.c.
• Bilinear – According to the bilinear property, the equation formed as follows: a.(rb+c) = r(a+b) + (a.c)
• Orthogonal – Two non-zero vector quantities are said to be orthogonal if a.b=0. The situation implies that the two vectors are perpendicular to each other, forming the 90-degree angle.

Conclusion

In a nutshell, the scalar product of two vectors is a medical operation of two different vector components. This method is applicable only for vector pairs that cause the same dimensions. Two main approaches define the functioning of the scalar product of two vectors. These are algebraic and geometrical definitions.

Another important aspect of our topic is the important properties of the scalar product of two vectors that define its main characteristics. The scalar product is commutative, distributive, bilinear, commutative, and orthogonal. The concept also follows other properties, such as non-associative property, where the scalar product between a vector and a scalar quantity is impossible.