Scalar multiplication

Introduction

In Physics, force, velocity, speed, and work are classified as scalar or vector quantity. Scalar quantities are physical quantities with magnitude and no sense of direction. Vectors have 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 multiplication and scalar multiplication. 

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

We must have a thorough understanding of scalar and vector quantities to understand scalar multiplication and its various components.

What are Scalar Quantities? 

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

Some scalar quantities are – volume, mass, speed, and time. 

What are Vector Quantities? 

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

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

Definition of Scalar multiplication 

The scalar multiplication of two vectors is the sum of multiples of their corresponding components. The answer obtained will always be a scalar quantity and a real number. 

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

We can understand this using an example: 

Let A and B be two vectors. When we multiply them, the scalar multiplication obtained will be as follows: 

A.B=|A||B| cos

Here, the angle between these two vectors is .

Upon expansion, this can also be written as: 

A.B=|A||B| 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, their scalar multiplication can be defined as a multiplication of the magnitude of A and the component of B on A.

How to Find the Scalar Multiplication of Two Vectors? 

The scalar multiplication of two vectors is calculated by multiplying the modulus of both vectors and the cosine of the angle between them. Simply put, you can find the scalar multiplication 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

The Formula for Scalar multiplication 

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

Algebraic Formula for Scalar multiplication

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

1. a. b = i =1naibi= a1b1 + a2b2 +……………+anbn

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

Properties of Scalar multiplication 

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

Property 1:

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

It is expressed in the form of the equation: 

x.y = y.x  

Property 2:  

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

Property 3: 

The dot multiplication 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 4: 

Two vectors are orthogonal when their scalar multiplication is zero. For example, vectors x and y will be orthogonal when x.y = 0. 

Applications of Scalar multiplication

The multiple applications of scalar multiplication make it useful for various fields such as game development and engineering.

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 = (A.B)/( |A||B| ).
5. Predicting the amount of power that solar panels can produce. 

Conclusion 

Vectors and scalars are concepts in mathematics that may be difficult to understand at first. But, with consistent study and comprehension, the knowledge becomes manageable. That was all about a dot or scalar multiplication and its different uses. From covering the different formulas of scalar multiplication to highlighting its properties, we delved deep into various aspects in this article.