VECTORS - Introduction and The Triangle Law of Vector addition

We all know we have units and quantities, without which the function of the world as we know it would cease to exist. (No I am not trying to be dramatic in my very first sentence )

Imagine you agreed to drop your friend to her house on your way to work

Friend - Wait! Take two here

You - ( looks at the cross road and stares back) Two what?

Friend - What?

You - Two kilometres, Two miles, Two metres, North , East , West ? (anxiously looks at your watch )

Friend - Just two

You- Get out right now

Some of the physical quantities have only magnitude, like measuring time, temperature, mass etc. These are called Scalars quantities.

While some require a specified direction Egs:- While hitting a coin in a carom, driving or accelerating a car etc. These are called Vectors, which will be the star of this blog

Definition of Vector

Vector quantities are the physical quantities, which have both magnitude, sense of direction and follows the triangle law of vector addition (we will be dealing with this later on in the same blog)

Egs: - Acceleration, Momentum, Torque, Weight, Velocity

Representation of Vector

Vector ‘P’ is represented as

( P in bold face or with arrow symbol on top)

Magnitude of Vector ‘P’ (i.e considering only the number and not direction) can be represented as

(unbold P )

And the direction of vector (now, remove magnitude) is represented as

( p with a cap)

Unit Vector

Mathematically (yes, this is a physics blog, I know, but Maths just can’t resist from poking its nose everywhere) Direction of vector, also known as Unit Vector is represented as

Orthogonal Unit Vectors

We learnt what unit vectors are, now it’s time to add a prefix to the same. Take a 3 dimensional figure, The unit vectors which are mutually perpendicular to each other and representing the axis of 3 – dimensional co-ordinate system is called Orthogonal Unit vectors ( Take the corner of the room you are sitting in, the length is X – axis, breadth is Y – axis and Height is Z – axis)

Consider a point C in this axis system. OC represents the positional vector of C (basically, measure the point from origin)

Vector C can be represented as:-

which is equal to (when we show magnitude and unit vector seperately) :-

Magnitude of the vector C is =

The direction of the vector ( according to what we defined unit vector , basically vector by its magnitude)

TRIANGLE LAW OF VECTOR ADDITION

Before we dive into the main concept related to this subtopic, Let’s clear something first

Consider two vectors; they can be either of these two ways

In the first way of representation (where both tails or head meet) , the angle considered will be the angle between the two vectors (interior angle) in the second way of representation (where tip meets the head of arrow) , the angle considered will be the exterior angle.

Now that that it cleared up, let us start the actual topic

The triangle law uses basic principles of plane geometry, It states that:-

The vectors to be added are drawn in such a manner that the tail of a vector coincides the tip of the preceding vector (Tip to tail). The resultant is defined by the vector drawn from the tail of the first vector to the tip of the second vector.

Magnitude of the resultant =

Inclination of the resultant vector to the first vector =

(Both these expressions have a long derivation explaining their relations, which will be covered in another blog soon, so keep your eyes peeled!)

And That is the triangle law of Vector Addition

If you are thinking, “Hey, Why would I ever use this ridiculously long expression with more alphabets than numbers in real life? I am pretty sure Sundar Pichai doesn’t go around using the word ‘vector’ every day and He is earning in millions”

Well, physicist really liked the language Latin hence named simple everyday things in a language no one understands (just to make the lives of students miserable, I mean what did we ever do to you physicist!)

If you played basketball, gone to the gym or are have acted as the Olympic weightlifting champion of your family by lifting a bag of rice over your head and yelled, “Ha! No one can defeat me” .... Basically thrown anything in your lifetime, you have used vectors.

To throw a basketball to your friend or teammate, you have to consider the acceleration of the ball along with the acceleration of your moving friend then aim it. Your *genius* brain calculates all this in a fraction of a second and relays the information to your arms (and we sadly steal the credit from our brain and tell it was our ‘gut feeling’) You are using the concepts of vectors even without realising it!