Basics of vectors

4 min readNov 26, 2020

Introduction

In physics, quantitatively analyzing different properties of an objects helps in many ways.

asking questions like:
How much is the speed?
What is the mass?
What is the height and weight?
gives us a number which helps us to quantitatively answer these questions. Such information is very useful to understand the object.

The number we get by asking such questions is what we call the Magnitude of a particular physical quantity. It basically answers the question that how much of something exist.

Many a times, only knowing the magnitude doesn’t help,
for example: you want to go to a classroom in your school campus which is around 100 meters from where you are, now just knowing the distance or the magnitude of length is not enough.
Information like a direction is essential to make sense of the magnitude or to answer the question ‘where should I go 100 m to find my classroom?’. To answer such questions knowing the Direction is essential.

Scalers

Physical quantities that convey enough information with just the magnitude are called Scalers.

Or, physical quantities that can be represented only by magnitude are scalers.

Examples include : volume, density, speed, energy, mass, and time.

Vectors

Physical quantities that in alone with magnitude cannot convey enough information and requires direction to make sense or to answer the question are called Vector.

or, any physical quantity that require to be represented with both magnitude and direction are called Vectors.

Examples include : force, velocity, acceleration, displacement, and momentum.

A directed line segment is the best example of a vector. Where the length of the line segment is the magnitude and the arrow represents the direction.

[ Note : in physics, vectors are something that we define. Meaning, something is not naturally a vector or scaler, instead we define them to be a vector or scaler.]

Position Vector

A vector that represents the position of a point in the cartesian axis is called the position vector of that point.

Like in the adjacent figure, consider a point P(x,y,z) in space with respect to origin O.

r represents a vector that describes the position of point P with respect to O.

Direction Cosines

The angles that a particular vector makes with the x,y&z axis are called direction angles. And cosine of these angles are called direction cosines.

Types of vectors

Zero Vector : A vector whose initial and final point coincide, or in other words a vector whose magnitude is zero.

Unit Vector : A vector whose magnitude is one.

Coinitial Vectors : Vectors with common initial point.

Equal Vectors : Vectors with same magnitude and direction.

Negative Vector : Vectors with same magnitude but opposite directions.

Addition of vectors

Vector addition is defined as the geometrical sum of two or more vectors as they do not follow regular laws of algebra. The resultant vector is known as the composition of a vector…………
Read from here..

Components of a Vector

In a two-dimensional plain, a vector can be divided or resolve into two different vectors in the x and y direction, the resolved vectors are called x-component and y-component of the vector respectively.
The components are the influence of a particular vector in the x and the y directions.

Learn how to find the resolved value of the components here….

Product of two vectors

When it comes to multiplying two vectors, we have defined two different ways of multiplication, scaler or dot product — whose product is a scaler and vector or cross product — whose product is a third vector.

Let’s have a look at both of them.

Dot/Scaler Product

consider two non-zero vectors a and b, the angle between them is θ.

Now, the dot product of vectors a and b is denoted by a.b and is defined as :
|a||b|cosθ (where |a|&|b| are magnitudes of vector a and b)
The product is a scaler.

Cross/Vector Product