Preloader
  • By Admin Koushi
  • (0) comments
  • March 15, 2025

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: 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 along x axis is 00)

Rsin00 = 0

Rcos300 = R

Rsin300 =

Rsin00 = 0

 

Rcos00 = R (consider angle of along y axis is 00)

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().

Admin Koushi

previous post next post

Leave a comment

Your email address will not be published. Required fields are marked *

contact info

subscribe newsletter

Subscribe to get our Latest Updates

Get updates On New Courses and News

© 2018 – 2025 Koushi All Rights Reserved