Addition of Vectors: Class 11 Physics Chapter 4 CBSE
Triangular Law of Vector Addition: Following the rule of parallel shifting, draw one vector and then draw another vector such that the head of one vector is joined to the tail of another vector, thereafter joining the tail of the former vector to the head of the later vector, we get the resultant vector of both vectors which is the addition of the vectors.
As an example, a displacement of 3 m east and a displacement of 4 m north, then find the addition of the vector
Let the vector = 3 m east and = 4 m north then their addition is
Hence the resultant of both vectors is 5 m northwest,so there are specific rule for vector addition,we can’t add them like the scaler quantity means the resultant can’t be 3 +4 =7
What is the resultant of two vectors when the angles between them is other than 90ยฐ
Let there be two vectorsย ย and ย then their addition is represented as follows
The resultant R of both vectors is given as
The vector sum of both vectors is not a simple addition,the method mentioned above is head to tail method,apart from that there are other methods of solving vector addition.
The resultant of head to tail method is applicable when both vectors are in a same line or perpendicular to each other,so when both vectors have a certain angle ฮธ between them then their resultant vecor is evaluated by parallelogram rule of vector addition.
Parallelogram Law of Vectors addition: In this method, two vectors are drawn tail to tail as they are adjacent sides of a parallelogram and then geometrically completing the parallelogram,then the diagonal obtained by joining the vertex of both vectors to the opposite vertex of the parallelogram shows their resultant.
Draw two vectors P and Q such that they join each other tail to tail
Complete the parallelogram ABCD, then draw a DEโฅCE
The sides AD =Q=AB,DC =P=AB shows magnitudes of the vectors
BDย will show the resultant of the vectors
Let both vectors have angle ฮธ between them and the resultant forms an angle of ฮฑ between themย and the magnitude of the resultant vector is R
Considering the right ฮBDE
โ ABC = โ DCE=ฮธ
BDยฒ = BEยฒ + DEยฒ
Rยฒ =(BC+CE)ยฒ +DEยฒ
Rยฒ = BCยฒ +CEยฒ +2BC.CE + DEยฒ
Rยฒ = Qยฒ + CEยฒ+ 2Q.CE + DEยฒ…….(i)
Let us find the value of CE and DE
Considering the triangle ฮDCE
cos ฮธ =CE/DC and sinฮธ = DE/DC
CE = DCcos ฮธ =Pcos ฮธ …..(ii) and DE =DCsin ฮธ= Psin ฮธ …..(iii)
Putting the value of CE and DE in equation (i)
Rยฒ = Qยฒ + (Pcos ฮธ)ยฒ+ 2Q.Pcos ฮธ + (Psin ฮธ )ยฒ
Rยฒ = Qยฒ + Pยฒcosยฒ ฮธ+ 2Q.Pcos ฮธ + Pยฒsin ยฒฮธ
Rยฒ = Qยฒ + Pยฒ(cosยฒ ฮธ+sin ยฒฮธ)+2Q.Pcos ฮธ
Rยฒ = Qยฒ + Pยฒ+2PQcos ฮธย
The above equation is used for determining the magnitude of the resultant of two vectors
Now
tanฮฑ =DE/BE
BE =BC +CE =Q+ Pcosฮธ
From equation (ii) and (iii)
tanฮฑ = Psinฮธ/(Q+ Pcosฮธ)
This equation is used to determine the direction of resultant
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