derive the expression for resultant of sum of 2 vector
Answers
Answer:
In summary, the resultant is the vector sum of all the individual vectors. The resultant is the result of combining the individual vectors together. The resultant can be determined by adding the individual forces together using vector addition methods.
Answer:
Here's the correct answer
Explanation:
Let P and Q be two vectors acting simultaneously at a point and represented both in magnitude and 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 vector. Then, according to parallelogram law of vector addition, diagonal OB represents the resultant of P and Q.
So, we have
R = P + Q
Now, expand A to C and draw BC perpendicular to OC.
From triangle OCB,
OB
2
=C
2
+BC
2
or OB
2
=(OA+AC)
2
+BC
2
. . . . . . ( i )
Intriangle ABC,
cosθ=
AB
AC
or , AC = AB cosθ
or , AC = OD cosθ
= Q cosθ [ AB = OD = Q ]
Also,
cosθ=
AB
BC
or , BC = AB sinθ
or , BC = OD sinθ
= Q sinθ [ AB = OD = Q }
Magnitude of resultant:
Substituting value of AC and BC in ( i ), we get
R
2
=(P+Qcosθ)
2
+(Qsinθ)
2
or , R
2
=P
2
+2PQcosθ+Q
2
cos
2
θ+Q
2
sin
2
θ
or , R
2
=P
2
+2PQcosθ+Q
2
R =
P
2
+2PQcosθ+Q
2
Which is the magnitude of resultant.
Direction of resultant :
Let ϕ be the angle made by resultant R with P . Then,
From triangle OBC,
tanϕ=
OC
BC
=
OA+AC
BC
or , tanϕ=
P+Qcosθ
Qsinθ
ϕ=tan
−(
P+Qcosθ
Qsinθ )
which is the direction of resultant.