The scalar product

In Physics, terms such as force, velocity, speed, and work are usually classified as a scalar or vector quantity. Scalar quantities only have magnitude but no direction. Vectors consist of both magnitude and direction.

Therefore, operations such as addition and multiplication can be easily performed on them. Multiplication, in particular, can be done in two ways, i.e., cross product and scalar product. 

While the cross product is a binary operation that gives a vector, the scalar or dot product, on the other hand, gives a scalar quantity as a result.  

In order to understand what a scalar product is and its various components, we must have a thorough understanding of scalar and vector quantities. 

What are Scalar Quantities? 

A quantity that has magnitude but no direction is defined as a scalar quantity and is often denoted by a number, followed by a unit. A good example of a scalar quantity would be – the distance traveled by a car in an hour or the weight of a bag. 

Some common scalar quantities include – volume, mass, speed, and time among others. 

What are Vector Quantities? 

A quantity that depends on magnitude as well as direction like force, displacement, momentum, velocity and acceleration is known as the Vector Quantity. It is denoted by a number and an arrow at the top (a) or a unit cap like a.

Example: Measuring the displacement or shortest distance covered by a car is Vector Quantity. 

Definition of Scalar Product 

When two vectors are multiplied together, the result you get is known as a scalar product. The scalar product of two vectors can also be described as the sum of products of their corresponding components. The answer obtained will always be a scalar quantity and as a result, a real number as well. 

Having widespread application in fields like engineering and astronomy, we can also calculate the scalar product by taking the product of magnitude of vectors alongside the cosine of angle between them. 

We can understand this using an example: 

Let’s say we have 2 vectors,  A and B. When we multiply them, the scalar product obtained will be as follows: 

A.B=AB cos

Here is the angle between these two vectors.

Upon expansion, this can also be written as: 

A.B=AB cos = A(B cos )=B (A cos )

Here it is clear that Bcos will be the projection of B onto A, while Acos will be the projection of A onto B. Therefore, the scalar product here can be defined as a product of the magnitude of A and the component of B with A or as a product of the magnitude of A and the component of B with A.

How to Find the Scalar Product of Two Vectors? 

The scalar product of two vectors is calculated using the product of the modulus of both vectors along with the cosine of angle between them. Simply put, you can find the scalar product easily by multiplying the magnitude & projection of the first vector onto the second vector. 

The formula for two vectors x and y would be: 

x.y = |x| |y| cosθ

Formula for Scalar Product 

Now that we’ve understood the formula for the scalar product of two vectors, let us take a look at the algebraic interpretations of the scalar product. 

Algebraic Formula for Scalar Product

In algebraic terms, the scalar product refers to the sum of corresponding entities in a series of numbers after being added together. The dot product for two vectors, a and b are as follows: 

1. a. b = i =1naibi= a1b2 + a2b2 +……………+anbn

Here Σ is the summation while n is the dimension of the vector. 

Properties of Scalar Product 

Now that you’ve got a fair idea of the formula used to calculate the scalar product, it is important for you to familiarize yourself with its various properties: 

Property 1: 

We can find the scalar product of two vectors by multiplying them to derive a scalar quantity which is also a real number. It must be noted that this concept applies to scalar products of two, three and more than 3 vectors. 

Property 2:  

The dot product of two vectors will always be commutative, implying that the sequence of non-zero vectors won’t matter, since the answer would always be the same.

 This can be expressed in the form of the below equation: 

 x.y = y.x  

Property 3:  

If x.y = 0, then either of x or y would be zero or cos θ = 0 that means  θ = π/2, which implies that either of the mentioned vectors are either 0 or perpendicular to each other. 

Property 4: 

The dot product is distributive. Implying that – 

x.(y + z) = x.y + x.z

x.(y-z) = x.y – x.z 

(x + y).z = x.z + y.z

x.(y – z) = x.y – x.z

(x – y).z = x.z – y.z

Property 5: 

Two vectors are orthogonal when their scalar product is zero. For example, vectors x and y are said to be orthogonal when x.y = 0. 

Applications of Scalar Product

Used in multiple fields such as game development and engineering, there are multiple applications of the  scalar product.

They are as follows: 

1. Search the shortest route to a destination  
2. Identify the total force applied in a particular direction 
3. Multiplying matrices in linear algebra 
4. Finding the angle between two vectors using the formula cos θ = (x.y)/(|x| |y|).
5. Predicting the amount of power that solar panels can produce. 

Conclusion  

This was all about what a dot or a scalar product is and the different ways it can be used. From covering the different formulas of a scalar product to highlighting its properties, we delved deep into a variety of aspects in this article.