Resolution of vector
Resolution of a vector into its components is the process to determining a set of vectors, whose resultant is the given vector. Each vector in that set is called a component of the given vector.
Prove: Let us consider 
(= 
) is resolute along the line OA and OB creates angles 
and 
with OC respectively. From point C, CD and CE are drawn parallel to OB and OA respectively. OD and OE represent 
and 
respectively.
From 
ODC, 
COD = , OCD = , then ODC = 1800– ( + ).
From trigonometry, we can write 
= 
= 
Or, = = 
Or, 
= 
= 
So, P = 
and Q = 
.
Resolution of a vector into two rectangular components:
Let us consider, OX and OY are two mutually perpendicular axes, where O is the origin. OB represents creates angle θ with +ve x axis. Let BA and BC are the perpendiculars on OX and OY axis respectively. and are the components of and represented by OA and OC respectively. I.e. = + .
The magnitudes of and are represented as P = R
[
= ] and
Q = R
[
= ]
Therefore, R = 
and tan
= 
.
Vector addition by resolution: Let us consider XX/ and YY/ are two perpendicular axes, where O is the origin. 
, 
and 
create angles 
, 
and 
respectively with +ve x axis.
The components of , and along X axis are respectively Px = Pcos, Qx = Qcos(
– ) = – Qcos and Rx = Rsin(
– ) = – Rcos.
The components of , and along Y axis are respectively Py = Psin, Qy = Qsin( – ) = Qsin and Ry = Rcos( – ) = – Rsin.
Let the resultant along X axis is Fx = Px + Qx + Rx (let Fx is along + ve x axis) and along Y axis is Fy = Py + Qy + Ry (let Fy is along + ve Y axis).
So, the resultant of Fx and Fy is F (= 
). If 
creates angle θ with + ve x axis then tan
= 
.
Example: 1. Find the resultant of three vectors OA, OB and OC as shown in figure, where R is the radius of the circle.
Magnitude of OA = OB = OC = R.
The components of OA, OB and OC along positive x and y axes are respectively shown in the table below.
vector | component along positive x axis | component along positive y axis |
|---|---|---|
| Rcos00 = R (consider angle of | Rsin00 = 0 |
| Rcos300 = | Rsin300 = |
| Rsin00 = 0 |
Rcos00 = R (consider angle of |
The resultant vector along x axis is R(1 + 
) and along y axis is 
.
The magnitude of resultant vector is R
= R
= R
If resultant vector creates angle 
with x axis then tan = 
= 
tan-1().
2. Determine the resultant of three vectors of magnitude 1, 2 and 3 whose directions are taken in the order of the sides of an equilateral triangle.
Three vectors P = 1, Q = 2 and R = 3 are arranged along the direction of the sides of an equilateral triangle. The angle between two vectors is 1200. Arrange all the vectors in XY plane such that is along positive x-axis.
The components of 
, 
and 
along x and y axis are respectively,
Px = PCos00 = P = 1 Py = PSin00 = 0
QX = QCos600 = 
= 1
Qy = QSin 600 = 
RX = RCos600 = 
Ry = RSin600 = 
The resultant vector along x axis is Fx = Px – Qx – Rx = 1 – 1 – 
= –
The resultant vector along y axis is Fy = Qy – Ry = 
– 
= – 
The result vector is F = 
= 
= 
= 
If result create angle with positive x-axis tan
= 
= 
= 
= tan300 Or, = 300
Resultant creates angle 1800 + 300 = 2100 with the vector of magnitude 1.
3. A force of 1000 N must be applied to a block in a particular direction. It is not possible to apply the force in that direction but two forces can be applied to 300 and 450 on either side of it in the same plane containing the given force. If the ratio of magnitude of forces is 
, then find the value of x.
Let the given force is F and F1, F2 are the two forces acting at 300 and 450 respectively on either side of it. The component vector of F perpendicular to F is zero. Hence, F1sin300 = F2sin450 or, 
Or, 
. Therefore, x = 2.
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