1.State parallelogram law of vectors. Derive an expression for the magnitude and direction
of the resultant vector.
Answers
Parallelogram law states that if two vectors are considered to be the adjacent sides of a Parallelogram, then the resultant of two vectors is given by the vector which is a diagonal passing through the point of contact of two vectors.
The addition of two vector A and vector B is resultant vector R = A+B+AB And R = A2 + B2+ 2AB CosΘ1/2And tan β = B SinΘ/ A + B CosΘ Where Θ is the angle between vector A and vector B And β is the angle which vector R makes with the direction of vector A.
Answer:
Answer
Parallelogram law states that if two vectors are considered to be the adjacent sides of a Parallelogram, then the resultant of two vectors is given by the vector which is a diagonal passing through the point of contact of two vectors.
In the figure
P
and
Q
are two vectors.with magnitudes equal to length OA and OB respectively and making angle θ between them. Complete the parallelogram, OACB,
Join diagonal OC , that makes angle α with vector
P
.
According to parallelogram law of vectors the resultant is represented by the diagonal passing through the point of contact of two vectors.
To find the magnitude of resultant , produce a perpendicular CD to meet OA produced to D.
From △ OCD,
OC
2
=OD
2
+CD
2
Now
C
D=
A
C sinθ=
Q
sinθ
AD=
A
Ccosθ=
Q
cosθ
O
D=
O
A+
A
D=
P
+
Q
cosθ
Putting these values and representing resultant vector OC by
R
, magnitude of the resultant is given by
R
2
=(
P
+
Q
cosθ)
2
+(
Q
sinθ)
2
=
P
2
+
Q
2
+2
P
Q
cosθ
In △ OCD,
tanα=
OD
CD
=
P
+
Q
cosθ
Q
sinθ
Resultant acts in the direction making an angle α=tan
−1
(
P
+
Q
cosθ
Q
sinθ
) with direction of vector P .
solution
