state the parallelogram law of vector addition and find the magnitude and direction of the resultant of two vectors vector a and vector B inclined at an angle theta with other what happens when theta equals to zero degree theta equals to 90 degree and theta equals to 180 degree
Answers
Let P and Q be two vectors acting simultaneously at a point and represented both in magnitude & direction by two adjacent sides OA and OD of a parallelogram OABD as shown in figure. Let θ be the angle between P and Q and R be the resultant vectors. According to parallelogram law of vector addition, diagonal OB represents the resultant of P and Q.
So, R=P+Q
From triangle OCB
OB
2
=OC
2
+BC
2
OB
2
=(OA+AC)
2
+BC
2
.........(1)
In ΔABC
cosθ=
AB
AC
AC=ABcosθ
AC=ODcosθ=Qcosθ
as [AB=OD=Q]
Also
cosθ=
AB
BC
BC=ABsinθ
BC=ODsinθ=Qsinθ
[as AB=OD=Q]
Magnitude of Resultant substituting AC & BC in equation (1) we get
R
2
=(P+Qcosθ)
2
+(Qsinθ)
2
∴R=
P
2
+2PQcosθ+Q
2
→ Magnitude
From Δ ABC
tanϕ=
OC
BC
=
OA+AC
BC
tanϕ=
P+Qcosθ
Qsinθ
∴ϕ=tan
−1
(
P+Qcosθ
Qsinθ
) .......(2)
When θ=0
o
ϕ=tan
−1
(
P+cos0
Qsin0
)
ϕ=tan
−1
(
1
0
)
ϕ=tan
−1
0
ϕ=0
When θ=90
o
ϕ=tan
−1
(
P+Qcos90
o
Qsin90
o
)
ϕ=tan
−1
(
P+0
Q
)
ϕ=tan
−1
(
P
Q
).
solution
