derive expression for magnitude of resultant of two factors using parallel of vector addition.
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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 .