Grade 11: Triangle and Parallelogram Law of Vectors Explained with Diagrams and Numericals AND EXAM BASED QUESTIONS

As all of us understand that in physics, we need to work with bodily quantities, which are called physical quantities. There are two types of quantities we set in our surroundings and they are :

1. Vector quantity
2. Scalar quantity

Now you might suppose what's vector and a scalar quantity are. Vectors are those amount who have both magnitude and direction and follows vector law of addition and subraction. For instance Displacement, velocity, acceleration and force are Vector quantity. But Scalar are those quantity who have magnitude but no direction. For time, density, work and current are Scalar quantity.

CONCEPT:(1)  Now you might possibly suppose displacement, pace, and acceleration are vector quantity and not scalar quantity and time, density, work and current are Scalar quantity why they are not vector quantum. It is because we all know that from the description of all three vector volume i.e. Displacement is how a way and in which course the object is moved likewise, pace tells us how rapid-fire and in which path object is moved and Acceleration is how presto is pace is changing in which direction. In these three Vector amount they have got both magnitude and direction so they are called as vector quantity but for density, work and current they are known as Scalar quantum because for Density we all know that viscosity is mass accoding to unit volume and we additionally recognize that mass and volume are scalar quantum so viscosity is also a scalar amount. For work, we know that work is made of force and displacement in the direction of force. Now, you might guess that Force and Displacement are Vector quantum then why Work is an scalar amount Because when two dot product of two vector are Scalar. Now for current, Current is the rate of flow of electric charge which means current have the simple direction form positive to negative but current doesnot follow the vector law of addition.

CONCEPT:(2) Now you might suppose what's Vector law of addition, when two vector are performing at the same time their overall which is called as Resultant can be found by placing head to tail.

VECTOR LAW OF ADDITION

There are notably 3 types of vector law of addition that are Triangle regulation of vector addition, Parallelogram law of vector addition and Polygon law of vector addition. However we are simplest going to speak about approximately  Triangle law of vector addition and Parallelogram regulation of vector addition.

TRIANGLE LAW OF VECTOR ADDITION

In Triangle law of vector addition when two coplanner vectors are represented by way of two adjoining aspects of a triangle taken inside the same order closing aspects taken from opposite order gives the sum of them is known as resultant vector or composite vector.

DERIVATION OF TRIANGLE LAW OF VECTOR ADDION

Let us bear on mind a triangle OAB and AMB with angle alpha and theeta. Now permit OA be 'a' vector and AB be 'b' vector and OB be 'r' vector. Now as all of us realize that Resultant vector is equals to 'a' vector and 'b' vector. Then from point 'B' and 'M' make immediately line and make factor 'M'. 

For Right angle triangle BAM

 

for angle "M"

cosθ=Base/Hypotenuse

cosθ=AM/AB

cosθ=AM/B   (AB is 'B' vector)

Bcosθ=AM

For angle "A"

sinθ=Vertical/Hypotenuse

sinθ=BM/AB

sinθ=BM/B     (AB is 'B' vector)

Bsinθ=BM

Now, By using pythagoras theorem

OB²= OM²+BM²

OB=(OA+OM)²+AM²

OB²= (OA+Bcosθ)²+BM²         ;(OM=Bcosθ)

OB²=(OA²+2OA.Bcosθ+Bcos²θ)+Bsin²θ      ;(a²+2ab+b²)

OB²=(A²+2A.Bcosθ+Bcos²θ)+B²sin²θ

OB²=B²(sin²θ+cos²θ)+A²+2A.Bcos   (taking common B²)

We all know (sin²θ+cos²θ)=1

HERE AND NOW,

OB²=A²+B²+2ABcosθ

Since, OB is resultant vector then,

R=√A²+B²+2ABcosθ

   

                                                                                                                 

Triangle law of vector addition is a basic conception used in physics and mathematics use to find resultant vector using two combined vectors. Placing head to tail forming an triangle whose another side represents resultant vector which can be find using Triangle law of vector addition.

Now, lets continue doing Parallelogram law of vector addition.

For parallelogram,

CONCEPT:(3)  The magnitude of the pass made of any two vector usually offer the vicinity of parallelogram.

Parallelogram law of vector addition state that if two vector as performing at the same time at a factor are represented each in route and magnitude by using two adjacent sides of parallelogram drawn from a point.

Now permit, POSQ be an Parallelogram then, OP be A vector and OS be B vector and OQ be the diagonal tested in figure 'i'. Now after making the slant in Parallelogram we will see that it absolutely became similar to Triangle law of vector addition. So, we are able to derive this much like Triangle law of vector addition.

For Right angle triangle OQN

 

for perspective "N"

cosθ=Base/Hypotenuse

cosθ=PN/PQ

cosθ=PN/B   (AB is 'B' vector)

Bcosθ=PN

For angle "P"

sinθ=Vertical/Hypotenuse

sinθ=QN/PQ

sinθ=QN/B     (AB is 'B' vector)

Bsinθ=QN

Now, with the aid of the usage of pythagoras theorem

OQ²= ON²+QN²

OQ²=(OP+PN)²+QN²

OQ²= (OP+Bcosθ)²+QN²         ;(PN=Bcosθ)

OQ²=(OP²+2OP.Bcosθ+Bcos²θ)+Bsin²θ      ;(a²+2ab+b²)

OQ²=(A²+2A.Bcosθ+Bcos²θ)+B²sin²θ

OQ²=B²(sin²θ+cos²θ)+A²+2A.Bcos   (taking common B²)

We all know (sin²θ+cos²θ)=1

HERE AND NOW,

OQ²=A²+B²+2ABcosθ

Due to the fact, OQ is resultant vector then,

R=√A²+B²+2ABcosθ

NUMERICALS RELATED TRIANGLE LAW OF VECTOR ADDITION AND PARALLELOGRAM LAW OF VECTOR ADDITION ASKED IN CLASS 11 BOARD EXAM.

• What is Triangle Law of vector addition.                                                                                           

Triangle law of vector addition state that if any two vector are performing at the identical time on a body represented by using magnitude and course via two facets of a triangle taken in identical order then the 3rd facet of the triangle represent both known as Resultant vector.

• What is Parallelogram law of vector addition.                                                                                   

Parallelogram law of vector addition state that if any two vectors are performing at the same time at a point represented with the aid of significance and route via two adjoining aspects of a parallelogram drawn from a factor the the slant of the parallelogram is drawn from the identical factor and the slant is referred to as Resultant vector.

NUMERICALS

• If any two vectors of magnitude 5N and 7N are acting at an altitude of 60 degrees the discover the Resultant using the triangle law of vector addition.

Soln:

A=5N

N=7N 

Given altitude=60 degree

 Then, now by way of the usage of triangle law of vector addition,

R=√A²+B²+2ABcosθ

R=√5²+7²+2.5.7.cos60

R=√25+49+35

R=√109

R=10.44N

• If any two vectors A = 8 N and B = 6 N are acting on altitude of 90°. Use the parallelogram law to discover the magnitude of the resultant.

Soln:

A=8N

B=6N

Given altitude=90 degree

Now, by way of the usage of the Parallelogram law of vector addition,

R=√A²+B²+2ABcosθ

R=√8²+6²+2.8.6.cos90

R=√64+36+0

R=√100

R=10N

At ultimate knowledge, the vectors Parallelogram and Triangle law of addition is very vital we are able to say that these are fixed questions in every exam. You can try these type of numericals and derivation for your upcoming exam build your confidence for your exam.

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